If you mean the player faction specifically, it generally doesn't do much on its own, that's what you're for.
If it's a commissioning faction, depends on things like their economic production (for generating invasion fleets; this part also relies on having enemies to begin with) and whether they're set to launch colony expeditions. Which faction were you working with?
Comissioning, roider union and mayasura.
I guess they both start of pretty weak with only 1 planet and so the more established factions just have a much easier time to start expanding and both these factions can easily get kinda screwed and left weak and vulnerable then yeah.
Can I for example support them economically by selling them everything I find and fulfilling planet requirements or such to boost them?
Also if I request a colonization mission from them it says its a "player" mission, does that mean I would gain that planet or would it colonize it for the comissioned faction?