I bought a brand new phone through Verizon do I get a year of Apple TV with that purchase?

The year offers from Apple ended per July 2021. The new offers are for 3 months now.

The offer can be claimed once. If you have redeemed before, then you can’t add a new free trial.

Read here how to redeem: Apple TV+ offer - Apple

Read here about the conditions: Apple TV+ offer - Terms and conditions - Apple

Read here if you have trouble redeeming: If you can't redeem your promotion for an Apple subscription - Apple Support

The current Apple TV+ offer from Apple is three months, with some limits.

For the Apple offer: Apple TV+ - Apple

Verizon might have a different offer, but you’ll need to check with them about that, and then how to redeem that.