Which power is exercised when state legislatures or national conventions vote for or against a constitutional amendment

The answer is popular sovereignty.

In overall, when both the Senate and the House of the U.S. Congress officially proposes an amendment by at least two thirds of votes, it later has to be voted on by the state legislatures or national conventions (called by the states legislatures). Only if the amendment is approved by at least 75% of the state legislatures, it becomes a law.

When state legislatures vote for or against a constitutional amendment, the popular sovereignty is being exercised as for this political doctrine holds that all political power (authorities of states and federal government) is created and sustained by the consent of its people, through their elected representatives (In this case, state senators).