Fedora 37 has removed support for VA-API accelerated encoding and decoding of H.264, H.265 and decoding of VC1 but here is another workaround.

Mihai Vultur has published mesa-git packages for Fedora 37 on Fedora Copr with full VA-API support. Installing them is fairly easy.

Open a terminal and run:

sudo dnf copr enable xxmitsu/mesa-git

to enable the mesa-git repo.

Then run:

sudo dnf update

to update all packages. 

That's it. Reboot the machine and enjoy accelerated video playback and encoding.