No they can't. That is what the Speech and Debate clause forbids. The precedent is already there (Brewster) that members of Congress are not except from enforcement of laws against felonies, of which bribery is one.
Any warrant application to search a Congressional office, must, like other warrants, provide probable cause that a crime has been committed, and must describe the things to be seized.
Your argument could just as well apply to any "fishing expedition" conducted by law enforcement. Oh they might be fishing, so they can't search. Well it doesn't work that way for private citizens, and in the case of felonies, treason and breach of the peace, it doesn't work that way for member s of Congress either.
They can however continue to avoid obeying the traffic laws, at least up to the point that they commit involuntary manslaughter by vehicle. Even then they have a track record of getting away with it, but not because of the Speech and Debate clause.
No one is disputing that "The precedent is already there (Brewster) that members of Congress are not except from enforcement of laws against felonies".
BTW a question: the search team refused to allow Jefferson's lawyer to be present during the search, (or the House Counsel or even the Seargent At Arms, which is probably what got Hastert so upset). Isn't one's lawyer usually allowed at the search if he's there and asks?