Guardians don't have any obligation to any organization to the City, sny mission we go on if one we choose to accept. So there isn't any "military" per say, instead there are multiple different factions and orders. The Vanguard don't own the Guardians (as displayed in this Infection outbreak and Guardian refusal to abide by the quarantine) but they do offer a lot to the Guardians, such as guidance, tips and tricks, powerful armor and weaponry and general mentorship. Since they're the first people to interact with young Guardians, they're seen as the main organization. But Guardians can do pretty much anything they want (Ex: Trinary Star Cult, Sunbreaker Order, Dead Orbit) and as a result a lot of different Warlock and Titan orders form. Hunters usually stay solo from that. So basically a conglomerate of freelance warriors. And there is a non-Guardian Army known as the Forces of the City