It is possible for your dog to get a tattoo but you should know that dog tattooing is somewhat controversial and might be even illegal: http://www.opposingviews.com/i/society/animal-rights/dog-and-cat-tattoos-piercings-banned-new-york.
Dog tattoo can also be done for identification purpose.
Tattooing dogs requires anesthesia which might be risky and that is why this should be done at the vet’s office in case it is needed.
In case you really want to get your dog tattooed you might consider temporary tattoos:
http://www.myfoxny.com/story/22189217/temporary-tattoos-for-dogs.