Yes, it is. See GitHub - pfmoore/runpip for a script that builds a pip zipapp.
But please don’t make hatch depend on pip. As I said, pip isn’t set up to work as a dependency of other tools, and you will get bug reports from people who try to upgrade hatch and it fails because pip can’t update itself (typically because the script wrapper upgrade fails).
Probably the biggest thing that has come out of this discussion so far is that I want to explicitly add a note that pip does not support being quoted as a dependency of other tools (because it has no supported Python API).