Can you be a part time real estate agent while on Active Duty?

Technically, the answer is yes. Your Command needs to approve your part time job, and there needs to be a very clear understanding between you, your Commander, and your employer that the military comes first - always. You may have to call out of shifts on no notice or even leave in the middle of a shift (although that is rare).

Because of the nature of real estate, where you schedule conforms to your client's and appointments are IMPORTANT, a job in real estate would probable not be a great option. But if you can make it work, and your Commander will bless of... go for it.

Technically, all requests of a 2nd job, moon-lighting etc. must be approved by your Commander, and it can in no way interfere with the unit's training/activities. So you schedule as showing/closing and the unit can't/doesn't let you have the time off due to requirements, and you're screwed. Keep that in mind.

Yes If you work for someones office it would need to be approved by command. If you don't have an employer, you may not even need approval.