need driver for hp laserjet P1108

If the HP Smart app doesn’t work, you’re headed for a replacement printer.

Preferably a replacement printer with AirPrint support, or with the related IPP/IPPS support for use from macOS, as that can avoid the need for vendor drivers.

You can usually connect an older printer to something else and print via that (HP JetDirect, an old Mac or PC, etc), but the “something else” can be more expensive, more problematic, and/or more hassles than is a replacement printer. (And if we do follow that path, we can learn more about printing than any of us might prefer knowing.)

if you decide to replace: Which Printer Should I Buy? - Apple Community

HP only supports that printer up to macOS 12 Monterey.

see > https://support.hp.com/us-en/drivers/hp-laserjet-p1100-printer-series/model/5099195

You can't because HP no longer supports that printer to work with new Mac's.

The options are to use an older supported computer or replace the printer with a new supported printer.