Dear Dylan,
I have exactly the same issue on my website.
Like you,I have installed Paid Memberships Pro 2.9, and WordPress 6.0.1, linked to STRIPE in test mode.
So I asked a friend (he is a developer) to install Paid Memberships Pro 2.9 on his website and to do the tests.
Same as me, when a client click on SUBMIT and CHECK OUT, the page stays the same.
No action.
My friend is looking inside the code to understand what is happening.
I would let you know, as I see, I am not the only one with this kind of trouble.
On the other hand, Paid Memberships Pro 2.9 works perfectly with Elementor Pro Archives.