Otto Jespersen, A Modern English Grammar on Historical Principles, part 5 (1940/1954) identifies "make do" as falling into a category he calls "Incomplete Infinitival Nexus-Objects." Here is his discussion of this construction in modern English:

In ModE the construction infinitive without S[ubject] is found chiefly in the common phrase make believe, which can take a content-clause or an infinitive with to, in modern American also an infinitive without to as the object of believe: [examples and subsequent cross reference omitted].

Rarer as in Williamson S 19 Di laughed too, but only to make seem as if she didn't care.

Thus also in the now frequent phrase make do (do in the same sense as in "That'll do"). As this is not found in NED, and in the Supplement only with one quotation (from a NP 1927), I print several literary quotations:

[William John] Locke [The Great Pandolfo (1925)] 143 if you can make do with hash and rice pudding, I'll be grateful | [John] Galsw[orthy, Flowering Wilderness (1932)] 630 always having to look forward to see if you can make do | [Somerset] Maugham [The Bread-Winner (1931)] 4.241 We shall have to make do with the family bus for a bit longer | [Dorothy] Sayers [Gaudy Night (1935)] 478 If I can't have the real thing, I can make do with the imitation.

Other cases with make are rare:

Lyly C 321 It is better to hate the things which make to loue the things which giue occasion of hate | Tennyson 417 Yet to be loved makes not to love again.

In Benson N 280 it'll make talk we may have the substantive and not the verb talk.

Thus, according to Jespersen, "make do" is a short form of "make to do," just as "make believe" is a short form of "make to believe." Jespersen doesn't mention "make due" because (1) "make due" isn't a short form of "make to due" and (2) given that he was writing in the late 1930s, he probably never saw an instance of "make due" in print where due wasn't functioning as an adjective in a longer phrase.

Although the earliest instance of "make do" that Jespersen cites is from 1925, in a novel by William John Locke, earlier instances are not hard to find. In fact, Locke himself uses it in another, slightly earlier novel, The Tale of Triona (1922):

To emotional girlhood the feel of money, money not to hoard and make-do for weeks and weeks with the spectre of want ever in attendance, but money to fling recklessly about, has its barbaric thrill.

The earliest Google Books match for "make do" that I found is from W. Percival, Do Without It: A Christmas Story (1864):

'Hist, child!' she went on, after seeming to doze for a little while—'it's little more than I have left you that you will want to make do—with Martha to help, who is a tidy, careful body—that is, if you manage right.

From Joseph Jacobs, "Lazy Jack," in English Fairy Tales (1898):

His mother was this time quite out of patience with him, for the next day was Sunday, and she was obliged to make do with cabbage for her dinner.

And from various articles in The Commercial Motor (1915):

Our leading Editorial deals with the new situation which has arisen owing to the use which the War Office is to make of steam wagons. Few alternatives are being left to the user of the heavier classes of vehicles than temporarily to make do with those of smaller capacity.

...

...of those owners who have taken our advice to have recourse to two-ton, 30-cwt., one-ton, and even smaller models, but we do not seek to ignore the unhappy case of those would-be owners who have lost their heavier vehicles and cannot make do with smaller ones.

...

There is a feeling, no doubt, with thousands of our countrymen enduring unspeakable hardships in the trenches, that we should all endeavour to continue to make do with such opportunities and means as come our way.

We might recount, were this the place to do it, many incidents which go far to show that our soldiers have had to make do with the wrong thing—with, for example, puttees which shrank under water to the point of stopping the circulation in the men's lower limbs, thus intensifying the agonies of frost-bite and gangrene.

In contrast, the earliest twentieth-century match for "make due with" that multiple Google Books searches turned up is from Michael Harper, "Seed," in Nightmare Begins Responsibility (1975):

When you said the universe / was a seed, / silence, / nothing at all, / that spirit-man / conjured it up / to make due with his life, / I only nodded, / and you said you were finished.

From Jewish Frontier, volumes 46–47 (1979[?]):

The small private hospital, operating with a skeleton staff, on that day lacked even a resident doctor, and had to make due with a few overworked, harried nurses and fewer interns who were unsure of themselves and unwilling to get involved in a case where the placenta had arrived and the infant had not.

From Bonnie Parkhouse & Jackie Lapin, The Woman in Athletic Administration: Innovative Management Techniques for Universities, Colleges, Junior Colleges, High Schools, and State High School Associations (1980):

On the other end of the spectrum are the women who rue the day GAA, cookies, and punch vanished, want no scholarships and recruiting, are content to make due with a meager budget, and try to provide sports for all candidates.

...

• Women should compete separately from men and make due with what they have instead of demanding more

And from "Varieties of Shows Offers Alternative to Kool Fest," in Billboard (July 18, 1981):

There was no extra financial support for the Soundscape Festival. Says Gillis, "We had to make due with our smaller contributions from such firms as Con Edison and Citibank."

It is indisputable that "make due" has occurred with some regularity in published writing for a long time. But at least in Google Books results, the overwhelming majority of those instances are part of longer word strings in which due functions as an adjective: "make due redress," "make due probate," "make due provision," "make due return," "make due entries," "make due inquiry," "make due allowance," "make due diligence," etc., etc. Only with the advent of "make due with" in Michael Harper's 1975 poem does "make due" armed with a seemingly nonadjectival due begin to appear in modern results.

The one clear exception to this tendency is remarkable for both its earliness and its solitude. From a letter to Henry Clay from Felix Grundy, dated January 13, 1827, reproduced in The Papers of Henry Clay: Secretary of State 1827 (1959):

Altho, this matter had given me so much concern, that to prevent all controversy, I actually offered a few days ago, to give my note payable in twelve months—for any sum, which Mr Erwin yr son in law should make due by a calculation, without then knowing or claiming any credit for the $2400, receivd by me in the fall 1817,— I never for a moment beleived, or suspected that you would do me an act of injustice—or claim one Cent more than you beleivd to be Justly due—and altho I have always used this language when I spoke of it.

Here "make due" seems to mean something like "establish as being owed and requiring payment by a certain date." It is an interesting usage, but not one that I have found anywhere else in Google Books search results.

Most style guides haven't addressed the issue of using "make due" in place of "make do"—perhaps because the usage is relatively recent and still rather marginal. But Margery Fee & ‎Janice McAlpine, Guide to Canadian English Usage, second edition (Oxford University Press, 2007) [combined snippets] has this entry:

make do, [make due] Make do means to manage in straitened circumstances (to make less do the job). It is probably mere inattention that causes writers to substitute make due for make do.

The Toronto Maple Leafs will have to make due without Owen Nolan for the next two weeks. CBC Sports Online 5 Jan. 2004.

(Here make due should be make do.)

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Update (November 15, 2020): Earlier instances of 'make do' from newspapers

Although the earliest Google Books match that I could find for "make do" is from 1864, an Elephind newspaper database search turns up matches from as early as 1849. From "Farmer's Corner," in the Boston [Massachusetts] Pilot (June 9, 1849), reprinted from Farmer's Advocate:

VINEGAR. Many families purchase their vinegar at a very considerable annual expense : some "make do" with a very indifferent article ; and others, for want of little knowledge and less industry, go without. It is an easy matter, however, to be at all times supplied with good vinegar, and that too without much expense.

The appearance of quotation marks around "make do" in this instance suggests that the writer was aware of the unusual idiomatic form of the wording. This same excerpt from Farmer's Almanac—with the same wording, "some 'make do' with a very indifferent article"—also appears in newspaper items in the Keowee [South Carolina] Courier (August 4, 1849), the Montrose [Pennsylvania] Democrat (August 16, 1849), the Lehigh [Pennsylvania] Register (October 25, 1849), the [Franklin, Louisiana] Planters' Banner (November 1, 1849), and the Staunton [Virginia] Spectator & General Advertiser (November 21, 1849). Notably, these subsequent versions of the item on vinegar vary certain details of in the wording of the item: one titles it "BEET ROOT VINEGAR," and another "TO MAKE DOMESTIC VINEGAR"; one changes "for want of little knowledge and less industry" to "for a want of a little knowledge and less industry," and others to "for want of a little knowledge less industry"; and one changes "a very indifferent article" to "a very different article." But all of them retain the spelling "make do"; there are no instances of this particular item with the spelling "make due."

Indeed, the earliest newspaper instance of "make due with" that a separate Elephind search finds is from a student newspaper at Kent State [Ohio] University published in 1960. Overall, Elephind reports 3,575 matches for "make do with" (from Australia, New Zealand, and the United States, from June 9, 1849, to December 19, 2018) and 97 instances of "make due with" (all from the United States, from February 12, 1960, to February 13, 2014).

Slightly earlier than the 1849 instances noted above is this unusual instance of reflexive usage from a letter dated May 23, 1847, printed in the [Brookville] Indiana American (December 17, 1847):

As soon as the storm ceased we look the flesh off the bodies, what we could make do us four days, and started. We traveled on six days without finding any relief: On the night of January, my husband gave out and could not reach camp; I staid with him, without fire; I had a blanket and wrapped him in it sat down beside and he died about midnight, as near as I could tell.

Also, from "Greenhouse in Winter," in the Belmont [Ohio] Chronicle, and Farmers, Mechanics and Manufacturers Advocate (March 18, 1853), reprinted from Horticulturist:

They [the writer's plants] began to look brighter, healthier, and grow and bloom better than my utmost care had ever been able to make do before. And now strangers always ask the same question when they see my plants, that I used to ask my neighbor. My answer is "Use the sponge."

From "Farmer Makedo's Excusion," in the Ottawa [Illinois] Free Trader (December 8, 1855), an article in which the character Mr. Easy Makedo learns lessons in proper farm management from his neighbor Mr. Premium Taker:

Makedo was in a quandary. He had not began to see what shift he could make do to go on with [after the axle on his wagon broke], when Mr. Taker came back, and insisted on his stopping for the night with him, saying he could put him on his wheels again at an early hour.

From "Wanted! More Labor or Less Land," in the [Montpelier] Vermont Watchman and Sate Journal (March 27, 1957), reprinted from the Rural New Yorker:

While one farmer raises eighty or one hundred bushels of corn on one acre, another raises from ten to thirty, and yet the soil may have been originally the same. But the one has put his labor into proper shape—he has sought to keep up and add to the fertility of his soil ; he has his pay for it. The other has more acres perhaps, but on the make-do principle, he has hurried over the preparation and culture of the soil, and all his ambition has been more the show of acres, than of full cribs.

And from "The Wheat Buyers Attorney," in the [Red Wing, Minnesota] Goodhue Volunteer (July 2, 1862):

There was but one [attorney] in town mean enough to do it [the wheat buyers' "dirty work"], and he didn't know enough to do it well. Nevertheless, as he had a good idea of his own ability, and was, withal, candidate for the judgship of this district, they thought they would make do with him. And inasmuch as he was interested in the swindling operation they thought he would put forth extra efforts.