Your best chance at doing so would be to reintroduce your father into her life. Don’t simply give his vaporeon back to him. Rather, make it a transitional period. Spend the next few days or weeks before you leave for Alola doing activities with your dad and his vaporeon so that his pokémon can get acclimated to life with him again.
I would also highly recommend communicating with your father. Explain that you’re concerned over vaporeon’s mental health and let him know that you’re worried she’s not taking your potential departure that well. As her rightful trainer, he should be able to come up with a plan to reassure his vaporeon and help her transition back to his team. He may ask to do some of her feedings and have her perform important tasks for him. Your job in these cases is to encourage her to allow him to do this.
It may be worth it to reassure her that you’ll be back as well. You can do this verbally up until you leave, but after you depart, it may be worth it to use each pokémon center’s videophone to call home and allow your father’s vaporeon to see and hear you—at first, anyway. These conversations, too, can happen more and more infrequently until it’s clear your father’s vaporeon will be fine with just a weekly or monthly call.
Ultimately, though, step one would absolutely be to communicate with your father. Remember that you aren’t his vaporeon’s actual trainer, so it’s up to him to come up with a plan to help ease the transition. However, it’s important for you to communicate with him and let him know what’s going on, and it’s also important for you to support whatever he does to ease his vaporeon back into his life.
Best of luck, anonymous.
The Pokédex According to Bill 