Moonlighting is usually subject to command approval. Most organizations will have standard forms/blank memorandum to request permission to moonlight. In the Army, the regulations for moonlighting say that it cannot exceed 16 hours per week, and must not interfere with the military clinical duties (e.g., poor performance due to lack of sleep after working an overnight shift). Furthermore, you cannot care for patients in your moonlighting job that are eligible for care in the Military Health System. As a military provider, your salary is paid to examine these people, so you cannot accept additional compensation from them in a civilian capacity. Some commanders will also want periodic reports on the amount of time spent moonlighting each week/month.
This is a general answer, and I am not sure if being a GMO places any special restrictions on you. As a resident, you are on training status, and thus more limited. However, as a GMO, I think you are considered "staff" and not subject to the restrictions of training status.
I moonlighted at a "Doc in the Box" minor medical center during my years as a Flight Surgeon. It is up to your local command structure to allow moonlighting activites or not. As has been said, you can't see military members or beneficiaries for a fee. Remember that you will also have to either be provided malpractice or obtain it on your own for moonlighting. This can get rather expensive and negate that extra income.
After only completing an internship, you'll be "qualified" to probably work at minor medical centers or staff small E.R. departments. Usually that amounts to covering weekend shifts so that the usual doc gets weekends off.
Moonlighting is not a problem as a GMO, command permission is required but is rarely an issue. The most common are the "Doc in the Boxes", they'll hire people with only an internship if you can get licensed in the state. Some people picked GMO tours based on their ability to moonlight in a certain state...about half require only one year of GME, others require more. Once you're licensed, you are allowed to practice "medicine and surgery", more than phlebotomy though insurance can be a problem. I know of people pulling down an extra 40k a year moonlighting while GMOs, though that's unusual. Around here, you can make about $40/hour in a DITB.
You cannot moonlight while in a DOD residency, their opinion is they pay enough and you need to be studying. I think that's fair.
I don't moonlight, I'd rather have the time off and I'm not a primary care kind of Doc, so my time as a DMO is plenty of patient contact. I like the active duty population, but beware of the civilians...they seem interesting when you're a med student, the adjectives change when you get out there.