They could, but that would be wrong. The official name of the decoration is the Medal of Honor. The private society is called the Congressional Medal of Honor Society, reflecting the tradition that the award was created in the name of Congress. The award is typically presented by the President in person, or if not by him, by the highest ranking available member of the chain of command. It is not presented by the Congress as a body or by any member of Congress.
Damn! Beat me to it! ...well done.
I might add, that if I were to be awarded the MoH, I would not accept it from this president, and I am the 7th generation of commissioned officers in my family.