If you get your senior pet from a reputable shelter, their basic vet needs have probably been met, but ask what has been done, and get the records if possible.
Either way, as soon as you can, get your senior to your personal vet so they can begin their relationship. My Chicago vet was a personal friend so I couldn't wait to introduce him to the new "older” member of my family. He could see the way Honey Bear and I had transformed each other right away. She had been checked out by her shelter vet, but I wanted my vet to double check everything and start his own records. Your vet is one of the most important people you introduce your senior pet to, so do it early and do it regularly for the good health of your pet.