Say a very corrupt president were able to use threats to make Congress do as they said and committed many impeachable offenses, but Congress wouldn’t get rid of them, could 3/4ths of the state legislatures remove him from office using an amendment?
There is perhaps no limit on the power of the states to amend the constitution. The process is this:
2/3 of the legistures of the states apply to Congress calling for an amendatory convention.
Congress shall call a convention (there is no time scale specified in the constitution, common law suggests that there should not be undue delay)
The convention meets, with delgates from each state, amendments can be proposed in advance, or (probably) raised at the the convention. An amendment "President <insert name here> shall be removed from office...." would suffice.
The proposed amendment is considered in each state legislature. when 3/4 of the states have approved it (either in their legislatures or in state conventions), it becomes part of the constitution.
This process has never been used. It would be a slow process. It is extremely unlikely that the 3/4 rate of approval could be achieved in a situation in which 2/3 of the Senate were unwilling to convict on an impeachment charge. So in practice this method of removing a President will never happen: It is too slow and too difficult.
The only part of the constitution that can't be amended is the "equal representation in the Senate" (although even that restriction is questionable.) Beyond that, the constitution is completely determined by the states. There is no check on this power.