Will storing a mix of vermouths alter their taste?
It would certainly be safe. Vermouth is a fortified wine, meaning that it is wine with added alcohol. The added alcohol gives the vermouth a much longer shelf life, even at room temperature after the bottle is opened. You can store open vermouth for at least several months without any discernible loss of quality. It will continue to be safe indefinitely as long as the bottle is kept closed, keeping the alcohol from evaporating.

That said, I can see no reason why two vermouths couldn't be mixed in advance. As a matter of fact, you could add the bourbon too. Perhaps some kind of alcohol super-taster could tell the difference, but I sincerely doubt that any mere mortal would be able to tell the difference between a just-mixed Manhattan and one that was mixed 3 months ago (assuming, of course, an airtight container).