In a Two-Headed Giant game, life gained by your teammate won't cause the ability to trigger, even though it caused your team's life total to increase.

If an ability triggers whenever an opponent loses life and causes you to gain life, such as the ability of Exquisite Blood, this will loop until either you win the game or a player takes an action to break the loop.

Because Vito's first ability doesn't deal damage, you won't gain life when it resolves if Vito has lifelink.

If you gain an amount of life "for each" of something, that life is gained as one event and Vito's first ability triggers only once.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time, Vito's first ability will trigger twice and you may choose a different opponent for each trigger. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, and/or planeswalkers at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.