Does the American bank which gave that person the home loan have the
legal right to immediately start foreclosure proceedings on that home
after the first missed mortgage payment, or is there a federal law
that prohibits American banks from doing this to a kidnapped person?
There is no federal law of this kind, nor am I aware of any state law of this kind (since foreclosures are governed predominantly by state law).
The Soldier's and Sailor's Civil Relief Act provides a process similar to the one you contemplate in circumstances where the home owner is in military service for the United States, but there are no other laws that govern someone in similar circumstances. Of course, the Soldier's and Sailor's Civil Relief Act would provide protections if the kidnapped person was in military service.
Generally speaking, the bank would not conduct any inquiry into why the person was not paying their mortgage.
But, starting a foreclosure after just one missed payment would be very unusual. Usually banks don't actually foreclose until three or more payments are in default, even though they have the legal right to do so after just one missed payment, and sometimes a state will have a minimum amount that must be owed and outstanding before a foreclosure proceeding of certain types can be commenced (this is $5000 in certain kinds of Colorado foreclosures).
In other words, would this bank have to wait until the kidnapped
person is either freed or declared deceased by the U.S. government
before it can begin foreclosure proceedings on the home?
In this situation, a more common option would be for an interested person, perhaps a family member, to seek a conservatorship for the kidnapped person and to either request that the bank refrain from acting as an act or mercy, or the conservator could marshal the assets of the person in question to pay the mortgage if there were sufficient funds available.