"Has he got all his vaccinations?" asked the dad, looking at the tiny puppy.

"Yes, and he has a very friendly personality," replied the shelter worker. "He doesn't have any behavior issues at all, just a lot of energy!"

Has he got is the correct question form for "have got" with a third-person singular subject ("he"). "Does he got" mixes two different grammar rules!

has is the correct third-person singular affirmative form of "have".

doesn't have is the correct negative form of "have" for a third-person singular subject. (If using "have got", it would be "hasn't got").