If I'm not mistaken the mission will unlock at Lt.Cmdr3 and you can either hail Starfleet to get the mission (forgot his name, but the same who offers the other exploration missions), or just zone in to the Hromi cluster and you'll be offered it automatically.

Personally I find these badges to be anything but worthless. It allows you to get green Mark iV weapons for your entire ship quite early on, with a few decent consoles to boot, if you start doing them at the earliest possible. The only other way to get a complete set of weapons is either through buying them on the exchange (at some 50k a piece) or with crafting tier 2 (but you still need to get the normal versions).

The only thing I do not get is the huge cost of the engines and shields. As I go for weapons first, I find that most of the time I already got a comparable item from a mission reward or drop before I would have enough badges.