My understanding is that the proofs that CH and not-CH are consistent with ZFC are both about ZFC and in ZFC. Is it possible to do these proofs about ZFC but in a weaker axiomatic system?
(It is also my understanding that there are multiple methods of proving "CH is independent of ZFC". I'm interested in the question of if there is any way of proving this in a weaker system, and I'm interested in either direction (ZFC+CH or ZFC+not-CH).)
These independence proofs only use elementary finitistic reasoning. Finitistic reasoning itself could be formalized as PRA (primitive recursive arithmetic). PRA is a much weaker system than PA let alone ZFC. The proof theoretic ordinal of PRA is just $\omega^\omega$. Most mathematicians regard statements provable in PRA as really true (with the exception of perhaps ultrafinitists).
In a sense, the independence results really show in a strictly finitistic and constructive way how one could transform a proof of $0=1$ from ZFC+(not)CH into a proof of $0=1$ from ZFC alone.
Some remarks about this can be found in the book Set Theory (2013 by Kenneth Kunen) (Chapter II.1 Informal Remarks on Consistency Proofs).