I don't think they have a vote, literally or figuratively. I think it is DoD only.
Interesting link here: http://www.medalofhonor.com/RecommendationProcess.htm
Apparently, the only place Congress would play a role is if it was outside the 2-year limit. (This is an Army-specific example, but I would imagine the other Services would follow a parallel staffing path.)
The following organizations and individuals play key roles in the Army Medal of Honor recommendation process:
CHAIN OF COMMAND
Submits award recommendation that meets the two year submission time limit to Department of the Army Personnel Command
MEMBER OF CONGRESS
Submits award recommendation that is outside the two year limit for submission to Department of the Army Personnel Command or the Secretary of the Army who forwards request to Personnel Command.
DEPARTMENT OF THE ARMY PERSONNEL COMMAND
Army Decoration Board - Merit Review, can disprove based on criteria (Cdr, HRC can overrule)
Senior Army Decorations Board - Recommends approval, disapproval, or downgrade.
MANPOWER AND RESERVE AFFAIRS
Concurs or nonconcurs with Board recommendation
CHIEF OF STAFF OF THE ARMY
Concurs or nonconcurs with Board recommendation
SECRETARY OF THE ARMY
Recommends approval or can disapprove. Also forwards packet to Chairman, Joint Chiefs of Staff for comment.
SECRETARY OF DEFENSE
Recommends approval or can disapprove.
PRESIDENT
Approves or disapproves.