Software Freedom Conservancy is likely the next org to take over the hole that the FSF will leave. It's already more or less a legal arm that has gone to court over license violations and won and a fiscal sponsor for free software projects. The big difference, I guess, is that they are not completely exclusionary of software that may be used on unfree systems or bring in unfree components, like how FSF has never once recommended Debian despite the fact that it's entirely free software in the base install.

Outside of that, there's various smaller groups that more or less just fiscal sponsors like Software in the Public Interest.