I can pay it on my phone bill correct

If you're asking about payment methods for the App Store and the iTunes Store – then no, Verizon Wireless is not "who needs to be billed."

If Verizon Wireless provides your cellular phone service, their bill is for your cellular voice, text messaging, and data service. It might also include buying the phone on the installment plan, or subscribing to certain services, based on their offerings and your contract with them. E.g., there is a Verizon plan that includes an Apple One subscription. I would guess that Verizon is your contact point if you get your Apple One subscription that way.

But I don't think App Store and iTunes Store purchases go through any of the phone companies (carriers).

Did you follow the steps in this related article to set up your payment info correctly?

If you need to change or update your Apple ID payment method - Apple Support