My 3 minutes of Googling this say that 34 state legislatures are required to call for a constitutional convention. The delegates at the convention draft the proposed amendment, and then it goes back to all the states where it must be passed by 38 states to become an amendment. The tricky part is Congress chooses whether it goes back to the state legislatures for ratification or to state conventions. A
state ratification convention would be a weird animal where the voters choose on election day the delegates who would attend the state convention.
So if enough grass roots level action happened, we could amend the constitution against the wishes of Congress.
Edit: Unless Mitch McConnell decides to not do his job like he did with the Supreme Court appointments. So I guess they can just let a proposed amendment die. There is precedent.