Is it grammatically correct to start a sentence with "Not a +noun"?

Is it grammatically correct to start a sentence with "Not a +noun"? For example, "Not a woman held a presidential cabinet position in the US until 1933, when Frances Perkins became secretary of labor." Is this sentence grammatically correct? WOuld native speakers prefer to say "No woman held a presidential cabinet position in the US until 1933, when Frances Perkins became secretary of labor." If both are acceptable, what are the differences? Thank you.

Starting a sentence with that phrase may be grammatically alright, but it is highly unusual in most contexts. For example, take this line from The Night Before Christmas (1823), where the construction serves as the subject to a verb phrase:

'Twas the night before Christmas, when all through the house

Not a creature was stirring, not even a mouse (...)

This sounds old-fashioned to most English speakers today, who would be more likely to say no one was stirring or no creature was stirring if they were not reading a Christmas story.

If you run a search for this construction, you're likely to see certain set idioms, like not a peep, which can appear at the start of a sentence:

But not a peep of concern can be heard. (source)

Not a peep was heard from the students, least of all my troublesome student. (source)

Outside of that, you'd be hard-pressed to find results for not a NOUN PHRASE. For instance, running a search for "not a woman _v" in the Corpus of Contemporary American English generated 45 results. Out of that list, none started a sentence, and most results were actually versions of whether or not a woman VERBS, e.g.,

If you want to use a similar construction for emphasis, not one NOUN PHRASE is more generative:

Not one woman has told her that she thought abortion was okay, she adds. (source)

Not one woman is visible among the hundreds of vendors except tiny "Annie," who peddles cigarettes at $3.50 a pack off her shopping cart, covered with old newspapers. (source)

Not one woman spoke to me at that party last night the whole evening. ("Oh Vey, You're Gay," The Nanny (1995, TV Show)

Otherwise, I would strongly suggest using "No woman held ..." which would be the most common usage.