Hi,
The HP OfficeJet 8010e All-In-One Printer is listed in About AirPrint - Apple Support as a model with support for the driverless AirPrint technology. An AirPrint-enabled printer is compatible with macOS (Big Sur). This means that you should be able to use this model without having to use/install drivers, provided that the printer and the Mac are connected to the same network (Wi-Fi router).
A printer manufacturer may have a special app for setup, ink management, and similar purposes. Some ink subscriptions may even require an active Internet connection...
However, AirPrint as such should work without additional software.
If there is a problem with AirPrint, and if all settings are OK, you may want to try the following complete procedure (in the exact order):
Power off the Wi-Fi router, the printer, the Mac computer, and all other devices/computers. Wait two to three minutes. Power on the router. Wait two to three minutes. Power on the printer. Wait a minute or two. Power on the Mac computer. Finally, power on all other devices/computers.
Add a printer on Mac – Apple Support (UK)
Reset the Mac printing system to solve a problem – Apple Support (UK)