Would Jane Austen have been influenced by the first wave of feminists such as Mary Wollstonecraft?
I believe that Jane Austen would have been influenced by Mary Wollstonecraft (1759-1787).
This would probably most especially been the case with Wollstonecraft's authorship of A Vindication of the Rights of Woman, with Strictures on Political and Moral Subjects (1792), a "radical feminist manifesto" that was printed even before Thomas Paine's Rights of Man (1791-1792).
Jane Austen, like Wollstonecraft, was a free-thinking feminist who questioned the place of women within society by flouting the conventions that limited the things women were "allowed" to do. However, Austen handled her "rebellion" in a much more conventional way. Where Wollstonecraft struck out on her own, took work to support herself and wrote under her own name, Jane Austen was secretive about her writing and when she published her pieces, her name did not appear on the cover.
There is no question, however, that Austen's female protagonists flouted convention and were heroic rather than vulgar or scandalous as one might expect some readers of her day to react. Born in 1775, Jane would have been approximately seventeen when Wollstonecraft's "call to arms" (A Vindication of the Rights of Woman, with Strictures on Political and Moral Subjects) was printed to address the inequities of society towards women. Jane wrote her first novel at the age of fourteen, so she would also have been well read even before Wollstonecraft's book was published.
And as a reader and educated woman, I believe it would be logical to assume that Jane would have found a kindred spirit in the person of Wollstonecraft, though whereas Mary would have flaunted the reading of such a book, Austen probably would have read the book secretly, just as she completed the writing of her novels.
Having both lived in England at approximately the same time, and both having been associated with a wide variety of free thinkers and/or writers, I would expect that both women, while not necessarily knowing each other, might have traveled in some of the same "sub-circles" of society, finding themselves in the "orbits" of some of the same literary figures of the day.
Yes. Jane Austen lived in the Regency English period (Napoleonic Wars), which was a literary period quite known for its female writers. It was actually an abundance of them that included Mary Shelley, Fanny Burney, and Anne Radcliffe. The possibility that she may have influenced by any of her contemporary and previously published female writers is quite high, as it is understood that she even took the phrase "Pride and Prejudice" from one of Fanny Burney's novels.
Therefore, to answer your question, it is a literary necessity to observe the styles of writing of contemporaries and non-contemporary writers, not to copy or take ideas from them, but because it is typical of any kind of artist to appreciate the themes and styles of others.