My guess is that the amount of antivaxxers that falsify documents in order to travel internationally will be small enough to not be a major factor. For one, it's going to be simply much easier to just get the damned vaccine. But can you imagine getting caught in a foreign country with falsified documents? Yikes!
Foreign governments will require it and even for the foreign governments that don't require it, the airlines that serve them might. Better plan on getting the vaccine if you're going to travel beyond the borders of the US.