I'm not a constitutional expert (but I did stay in a Holiday Inn a while back), but I think a request for a vote for censure can be brought to the floor in a deliberative body by any member at any time during normal business once the member rises and is recognised by the presiding officer or "chair" in the case of the U.S.House of Representatives.
Censure itself has no standing or reference in the U.S. Constitution so far as I know, so it would be a procedural and symbolic affair, allowed by standing rules of the House, and only meaningful if agreed to by a majority of those voting on a censure motion.
It would in essence be the equivalent of a public reprimand. There is no penalty to the member who is censured, other than public embarassment.
It's a little more serious than a slap on the wrist, I suppose.