Can a 501(c)(7) receive external donations from non-members?

The IRS allows some income to be from non-members, but they do limit the outside income.

Effect of Nonmember Income on Exempt Status

A social club may receive up to 35 percent of its gross receipts from nonmember sources, including investment income. Within the 35 percent amount, no more than 15 percent of gross receipts may be derived from nonmember use of club facilities and services. Where the permitted levels of nonmember income are exceeded, all facts and circumstances will be considered in determining whether the club continues to qualify for exemption.

Unrelated Business Taxable Income

The Internal Revenue Code provides special rules for calculating the unrelated business taxable income of social clubs that are tax-exempt under section 501(c)(7). Under these rules, clubs are generally taxed on income from non-members who are not bona fide guests of members. The fact that income derived from non-members is used by an exempt organization in furthering its exempt purpose (such as expanding the club's facilities) does not change the fact that the income is from an unrelated activity. Clubs are also generally taxed on income from investments.

So if in your case "So she makes a large donation to them so that they can build a clubhouse." The person would have to be a member or they could jeopardize the tax exempt status for the organization. If your club is large enough the gift wouldn't impact the tax exempt status but the income would still be taxable.

If the person isn't a member you would have to make them a member before the gift is received. They would have to meet your qualifications for membership.

I guess a workaround would be to give the donor a special non-due-paying membership type, in exchange for the donation?

IRS Pub 557 discusses this idea:

The fact that a social club may have an associate (nonvoting) class of membership won't be, in and of itself, a cause for nonrecognition of exemption. However, if one membership class pays substantially lower dues and fees than another membership class, although both classes enjoy the same rights and privileges in using the club facilities, there may be an inurement of income to the benefited class, resulting in a denial of the club's exemption.

So if the idea is that this person pays 100K in "dues" and everybody else pays only 100 in "dues" that would concern the IRS.

If the gift is large, consult your tax advisor.

Also this isn't a tax deductible donation, because your organization is a non-profit your aren't a 501c(3) charity.