The states cannot act to implement such a manner unilaterally, and if they did, it would be deemed unconstitutional. It would have to be through Constitutional Amendment, which the states would then ratify.
Having to wait 2 or 6 years to rid a member who effectively lied their way into office is unacceptable. Even the Founding Fathers expected that (for U.S. Senators at least), that when the legislatures changed hands (which in those days were often held yearly), said Senator would resign immediately and allow for the Governor to appoint an interim Senator pending the election by the legislature for a new Senator.
Since that was considered a gentleman’s agreement and was not ratified into law (or as part of the Constitution), U.S. Senators recognized they weren’t bound to abide by that and would hold on to their seats for the full 6 years in the hope that things would be more favorable for them when they came up for reelection.
I think that has become a grievous mistake, and has only empowered the political left in this country (in both parties). If members in both Houses were accountable from day one if they were not doing the people’s business, it would drastically alter their behavior. You wouldn’t have 25 RINOs holding the House hostage in doing the bidding of the Demonrat opposition when swift recalls would be held to remove them.
You are correct that the Constitution does not permit states to recall their House and Senate elected representatives.
That does leave a Constitutional amendment as the only means for having recall elections against sitting House of Senate members. That is a strange hole let if the Constitution because the House and the Senate are able to “expel” members with a 2/3 vote, yet their own constituents cannot do as much.