While we can't know for sure, I would say it is highly likely that Jane Austen read and was influenced by Mary Wollstonecraft. For evidence of that, I would point the reader to Miriam Ascarelli's 2004 piece in Persuasions discussing feminist connections between the two writers (I would object, however, to Ascarelli's characterization of Vindication as strident and rambling).

While Ascarelli notes similarities between Austen and Wollstonecraft in terms of women's education, Ascarelli also points to actual links between Wollstonecraft and the Austen family:

Austen biographer Claire Tomalin offers some convincing biographical evidence that Austen is likely to have known of Mary Wollstonecraft and her work. She notes that Sir William East, the father of one of George Austen’s former pupils, was a benefactor of Wollstonecraft. Furthermore, Sir William was a neighbor and friend to Austen’s uncle, James Leigh-Perrot. After Wollstonecraft attempted suicide in 1796, Sir William was credited with being particularly kind to her during her recovery. While this does not specifically link Austen and Wollstonecraft, it makes it plausible that the Austen family knew of Wollstonecraft and her ideas.

Beyond that, Austen was an extremely well-read woman. Her novels also show the troubles that a lack of good education can cause for a woman, such as Emma Woodhouse. Further, as Ascarelli and other critics have pointed out, the period in which Austen was writing was one of censorship. England has never had a First Amendment guaranteeing freedom of speech. During a time of war with France and heightened sensitivity to the idea that radicals might be secretly helping the French, writers had to be very careful what they said. Since people like Wollstonecraft were in disrepute, Austen would have had to quietly weave a critique of women's education into her novels, which she does.

This is a good question. In terms of dates, it is not impossible that Jane Austen would have heard of Mary Wollstonecraft or encountered some of her work. On the other hand, there are several factors that make direct influence improbable.

For 21st century readers, both these writers fall into the category of late 18th/early 19th century female authors, and both may be studied together in classes on women’s literature and often read from a feminist perspective.

From a 19th century perspective, Mary Wollstonecraft was a scandalous woman who had two illegitimate children. Respectable women would not associate with her or read her work. Jane Austen was a clergyman’s daughter and outwardly conventional. In Northanger Abbey, we get a sense that although she sees a single young woman riding in an open carriage with a man on Sunday as more imprudent than immoral, she thinks such behaviour best avoided. We don’t see evidence that she approved of any radical ideas (or of radicalism as a mode of action) , but like many latitudinarians, would have argued that better education for women (e.g. reading Addison and Blair rather than Anne Radcliffe) would make them better members of society and better Christians.

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.

Would Jane Austen have been influenced by the first wave of feminists such as Mary Wollstonecraft?

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.