Currently, the only way of getting Vanguard Rep is to complete Vanguard Bounties, or to complete the exploration missions. The rewards for both of these are minuscule, and a great deal of grinding is required to advance your reputation level. Like, hardcore MMO levels of grinding. Every exploration mission besides the target elimination missions only give +10 Vanguard Rep, and nothing else. It takes 2000 points to level from 1 to 2. Assuming you don't complete bounties or the target elimination missions, it would end up taking you a comfy 200 missions to advance one rank in this system. Thankfully, the bounties and target elimination missions provide a little more reputation, but there's only 3 bounties per day which give you an average of 200 rep overall, and the target elimination missions only give 25 rep, 100 or so Glimmer, and about 10 Spinmetal. If I haven't made it clear enough already, it's insanely tedious to get Vanguard levels with the current system.
After several hours of playing these exploration missions, they become extremely repetitive and unrewarding. Oh, go stand over there for 10 seconds. Oh, go stand over there and press x for 5 seconds. Go shoot x enemies, and then salvage the induction rods out of their corpses. I actually ended up doing the same exact mission twice in a row on two separate occasions. I just can't analyze or scout hard enough, apparently.
I would assume that in the final release there will be more ways to get rep for this faction, but nonetheless only getting 1/200th of the way there after a mission is ridiculous.