Is it acceptable to begin a declarative sentence with "Am"?

I think you need to say "I am...." or your sentence is missing a subject. In other languages, such as Greek, you might say "Ego..... " but that translates to "I am;" in that case there is an understood first-person subject caused by the conjugation of the verb "to be."

This is not, however, acceptable in English.

There is a style of English speaking which drops subject pronouns, which I associate (perhaps wrongly) with Colonel Blimp-type figures. Something like:

Went up to town this morning. Met Caruthers at the club. Just got back from Africa. Ate a well lubricated lunch together for old times' sake. Am now coming home by train.

It is not standard English, and is prone to ambiguity. But it exists.

Scottish English has a common replacement Am for I am:

Am going to be late tonight.

This can be compared to the contraction of the negative form, which doesn't drop the I:

I amn't going to be late tonight.

I'm not sure in what context you're seeing people use this, but I would guess that it's internet or texting related. If so, then it's just typical modern-day internet prose (i.e. the continued downward spiral of the English language).

Technically, it is not grammatically correct. A sentence should have a subject and a verb (at least that was how I was taught). I certainly wouldn't use it in any formal writing.

It’s quite normal to drop the subject pronoun in contexts where brevity is important. It is certainly a practice found in emails and texting, but it was previously found in their predecessor, the telegram, and many people drop the subject pronoun when taking notes. It is also dropped in quite formal prose to avoid repetition, as in, for example, I went to New York last week and am going again tomorrow. It is a form of ellipsis, and should not be considered ungrammatical.

I believe some people find it easier to (only) use "am" in cases where they have already used the subject pronoun and have to use it a second time in the same sentence. Example: "I went on maternity leave on December 1st, and am not returning to work." ...It would just save you the having to specify that YOU did something, and (again) YOU will (or are) doing something as a consequence of. - However, (to me) it would feel more (grammatically) correct to use the subject pronoun as many times as you have to. Example: "I went on maternity leave on December 1st, and I am not returning to work."