Don't know if you're the one being helped by the service
dog, or if it is you who handle a service dog...
So, briefly, within the US you should not encounter difficulties. For foreign countries, you should write to the ones on your itinerary directly to ascertain what would be necessary for you to do (and some may not have anything) to avoid the quarantine periods that are standard.
Get creative. Find out what is their biggest concern. For instance, there is no rabies in New Zealand
. They don't want to import
it. But your vet could tell you if there is a blood test your pet could submit to for it...
I know seeing eye dogs
can travel on airplanes, but I don't know what happens offshore
with them. Sorry not more help.