Vulkan don't show up in the "Backend" list.

[LunaMoo]

Vulkan backend is only available for windows and android currently, linux support will definitely come someday, but unless you want to implement it yourself(which will likely involve updating SDL and Qt before anything else), you will really just have to wait - probably 1.6.0 will have it ~ sometime next year.

Edit: If you have a github account, you can check https://github.com/hrydgard/ppsspp/issues/9031 and click "subscribe" on the right to be spam.., ekhem informed on email with anything related.