When a weapon has the "Ensnare" listed, when is it activated? Upon a success?

Weapon qualities, like Ensnare, are activated on 2 Advantages, unless otherwise specified. So you'd need a success plus 2 Advantages.

What if it was a successful roll, but due to the target having a high soak, no damage was inflicted to the target, but you also rolled 2 advantages. Would you still be able to activate "Ensnare"?

You're good. As long as the roll is successful you don't have to bypass soak with damage.