In terms of the Marvel Cinematic Universe movies, no specific Avenger has been shown to own a pet dog. There is Captain Marvel, who owned a pet alien cat named Flerken.
However, the Hawkeye TV series featured a famous dog character taken from the comics. This would be Lucky, the Pizza Dog. Hawkeye, also known as Clint Barton with his superhero getup taken off, meets Lucky. He becomes smitten and decides to name him as such to make the dog his lucky charm. Lucky then forms a tight relationship with one of Clint's protégés, Kate Bishop. Throughout the show, they travel together often.