Some Star Trek Original Series headcanons:
Kirk- Stormcutter. They're clever, proud, skilled leaders, and have a bit of a playful streak, much like the famous starship captain
Spock- Silver Phantom. Mysterious, elusive, and serene, but when they do fire, it's incredibly hot and explosive. Feels like a good metaphor for a Vulcan
McCoy- Buffalord. Being a self-proclaimed "old country doctor" who's none too thrilled about being dragged into the dangers of space, I'd say he could easily find a kindred spirit in a cattle-like healing dragon who does NOT like to leave their comfort zone
Uhura- Dramillion. If Hiccup's flaming sword is any indication, fire can be a powerful tool to communicate and connect with dragons, so a species that can perfectly mimic others' fire seems like a fitting parallel to a skilled linguist in charge of the ship's communication
Scotty- Shovelhelm. Friendly dragons who love to build things and find creative solutions to problems, having one as a friend on the ship would be a great help to the "miracle worker" engineer
Sulu- Changewing. They're curious and love to copy what they see, and Sulu is also known to pick up a lot of different hobbies. One of his more prominent hobbies was his botany that involved strange and somewhat dangerous plants, and Changewings are skilled at using their camouflage and leaf-frills to disguise themselves among nature, so I could easily picture him taking in a "new plant species" that turns out to be an acid-breathing dragon
Chekov- Scuttleclaw. The youngest among the main cast, he sometimes runs into problems due to inexperience, but he's energetic and talented in a way that can often prove an asset as well, making the young dragons described as a "hyperactive powerhouse" a good fit
Kirk from Star Trek would ride a Stormcutter!
Spock from Star Trek would ride a Silver Phantom!
McCoy from Star Trek would ride a Buffalord!
Uhura from Star Trek would ride a Dramillion!
Scotty from Star Trek would ride a Shovelheim!
Sulu from Star Trek would ride a Changewing!
Chekov from Star Trek would ride a Scuttleclaw!