This probably a stupid question but do solar and batteries depend on rare earth metals and their supply?

The quick answer is yes, today. But there are battery technologies that require less and less in development.

Also, rare earth elements are not that rare. But they are not concentrated, and finding concentrations of them is kinda rare. Event then, you have to mine a lot of area to get them, which is not great for the environment. And since Americans (and everyone ex-China) has not been doing it for decades, only China has advanced the technology to extract and refine it for decades.

This lack of refining is similar to our lack of working on solar which will but us behind potentially forever, or until there is a big enough disruption to overcome the decade of experience. You can look at chipmaking and see that such things are not easy.

The answer depends on the kind of battery chemistry and how literally you mean "rare earth". If you take some slack on the definition and just mean "metal stuff in limited supply", then many battery chemistries have limited supplies.

There are, however, some chemistries with really nice supply chains. The Iron Redox Flow Battery (IRFB) really only needs iron and iron chloride as reactants. Those batteries are being commercialized, but they aren't common (yet?).