> Is this something that congress could (technically, not politically) fix themselves, or do they really depend on a nomination from the executive branch?

You could have any vacancy for more than X-days be filled by the minority party. (i mean even nominating an acting member -- you don't need to get an approval)

Proposal assumes two parties and would codify this. Two parties is one of the ways Americans keep getting themselves screwed. No need to codify it. Maybe one day we shall fix it?

Other parts of norms and conventions in USG assume two parties as well.

They can't appoint people, but they could change the quorum rules to allow the currently sitting commissioners to regulate. It would require Presidential signature, though, or enough votes to override a veto.