Hey everybody,
In my country (Canada) it is more affordable to finance a phone through a carrier (Fizz Mobile) than it is to purchase a phone from a store. Years ago my carrier financed cell had preloaded apps baked into the firmware, and it made me wonder about their data practices.
If I finance my phone through Fizz Mobile, instead of buy it at Best Buy, is there a privacy difference? I will be using GrapheneOS of course. 
Negligible difference, if any.
One difference is which you consider more likely to have a databreach leaking customer buying records. Which happens all the time to online stores, for example canada goose had a data breach leaking customer order info among other stuff. Have I Been Pwned: Canada Goose Data Breach
For the actual phone, as long as it is factory unlocked there should be no difference.
I would mainly check the non-privacy practical bits before choosing the cheaper option: is it factory unlocked, is OEM unlocking available, and can you return it if GrapheneOS install requirements are not met. If those are fine, the installed carrier apps should not matter after you flash GrapheneOS.
The privacy difference is more likely in the purchase/financing records than in the phone itself. Financing through the carrier may mean another account, credit check, or longer billing trail, so I would factor that in with the price difference.