I gave my dog external deworming today but not internal, and it got its first vaccine the same day; will there be a problem?

If the dog is healthy and a check shows no internal parasites, vaccinating without internal deworming is generally fine. But the normal order is internal deworming a week before vaccines, since internal parasites cause gut reactions that affect vaccine efficacy. External deworming is usually drops; with a reliable major brand, it can be done together with vaccines and has little effect on the dog.