Possibly, if non-essential administrative personnel are furloughed. USNA is a non-operational training command, far back in the DOD priority queue behind operational units.
Plans for this eventuality will have been in place for some time; essential and non-essential roles, functions and staff will have been designated. Mids will get fed; some admin functions will not be done by the usual people. Military profs, even if unpaid on the next payday on the 1st, will pick up the slack for civilian profs placed on furlough. Ditto other jobs.
In Admissions, there may be slowing of updates, data entry, etc., because of fewer staff available. They have no choice. Even if civilians wanted to come into work, if placed on furlough, they may not.