Can a NCO make a Soldier use nair to remove facial hair?

There are specific facial hair products. Nair is for legs and/or body hair

Lol, shaving bumps = a shaving chit.

No.... but yes (ish)
If a Soldier does not have a valid profile, then an NCO CAN require the Soldier to present a clean-shaven appearance. The NCO cannot mandate HOW that is achieved.

Short answer: no. NCOs do not have the authority to make you use any particular commercial product. They can however order a soldier to comply with grooming and appearance standards without specifying the method used. If the soldier's appearance is unacceptable despite making an honest effort to comply the NCO may recommend the soldier try other methods. The moment that recommendation takes on the character of an order to use a specific product, the NCO has stepped out of line.
Warning: saying this to the NCO is not likely to result in a positive outcome for the soldier. Instead, the soldier should tell the NCO he is making every effort and he would be grateful if the NCO could help him get that appointment with the medics.

Nair is a bad idea to be used on the face. Razor bumps can be grounds for a shaving profile. Nair is unnecessary.