'Were' and 'Should' are not interchangeable grammatically. 'Should' is a modal verb and is followed by the bare infinitive. The verb to be (was/were etc) can never be followed by a bare infinitive.
Your sentence should either be:
Should the President and Vice-President of the United States become incapacitated,...
or
Were the President and Vice-President of the United States TO become incapacitated,...
The second construction is [be] + [to] + infinitive. Both sentences describe an unlikely possibility in formal language. There is no significant difference between them.