Absolutely! Service pokémon, for example, are always separate from your full team of six, as if you have one, then they’re absolutely necessary for your day-to-day life. It wouldn’t do at all to force you to have one less pokémon to battle at your side, just because you need another one for medical reasons.
Outside of service pokémon, people often keep pets in addition to pokémon reserved solely for battling. These trainers are often more domestically-inclined, meaning the reason why they have no problem keeping pets is because they stick close enough to home to take care of them. While teams can double as pets, some people find it’s easier to pamper a pokémon that you’re not trying to train for hard battling. There are no limits to the number of pokémon you can keep as pets in your own home, but for obvious reasons, you can’t take many of them with you on your journey. Typically, the rules for “companion pokémon” (or those that are basically pets or team mascots, rather than team members) varies from region to region, but most of the currently recognized leagues only allow for one companion per trainer. This is largely for similar reasons to the six-pokémon rule: any more than that, and it may get difficult for a trainer to manage.
In other words, you absolutely can bring your umbreon along, so long as he’s properly registered as a pet or companion, rather than as a battler. You can do this by visiting your local pokémon center (or wherever you register for a league) and informing them of your umbreon’s status.
Of course, keep in mind that the rules are extremely strict about using your umbreon to battle, and once he’s registered as your pet, you can’t even use him to help you catch new team members. This may work out for him if he continues to dislike battling, but should he change his mind, you will need to visit a pokémon center to adjust his status.
The Pokédex According to Bill 