Can I purchase v-bucks on my mobile phone then have it transfer to my switch too so while using the same linked account on both.
If you use the same account on your phone as on your Switch, it would transfer as it's both on your account. Gameplay and any user data should be the same; the only thing that differs is the platform you play on.
You have to have both accounts linked.
First, if you haven't already linked your Nintendo account with an epic games account, you need to do that first, to sync your progress from mobile, or the other way around. Then, once you purchase the V-Bucks, they should transfer from one place to the other.