I doubt very much that such an attack would paralyze our ops. POTUS and VPOTUS are only the top dogs that issue orders, but orders can come at several levels below them. No one is going to just sit around and wait for someone at the new top echelon to hand down directives. Military ops get their marching orders from executive staff, and don't need Pentagon to give them the go ahead on such things as defensive adjustments and activations, launches, counterattacks, patrol mods etc. Tac Ops don't suddenly stop just because the top dog gets eliminated, we don't have that sort of command and control structure.

Likewise, most of the commerce and civilian ops would continue to be governed at the local level. Intrastate and interstate commerce/transport would continue to be run by the state governments and local municipalities. Some mass transit might be affected, and utility controls might be adjusted from the federal level, but again, this doesn't require top level executive management. Plans are already in place to deal with such contingencies, so decision making can stay at a lower, more regional and responsive level.

An attack on DC would not put us in the sort of paralyzing chaos that puts our country in jeopardy. It would create some problems, but we can't be shut down that easily.

There will have to be some adjustments made eventually, but the nature of our society pretty much negates the intended immediate affects of an attack on our central government.