[Chemistry] Why is it impossible to add or remove protons to an atom?

Who's telling you it's impossible?  it happens all the time.  Ordinary chemical reactions involve the rearrangement of electrons and do not involve changes to nucleus of atoms.  Protons (and neutrons) are nuclear subatomic particles and are therefore not changed in ordinary chemical reactions.

However, we have nuclear events that do change the numbers of protons / neutrons in atoms.  Fusion, Fission, decay (alpha, beta, gamma, etc), etc.  These events are much higher energy than ordinary chemical reactions because of the "strong forces" within atoms.

It's not impossible, just hard to do and thus relatively rare.

https://en.wikipedia.org/wiki/Proton_emission