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.