The core of this idea is that a species has access to some sort of gate technology, which allows them to cross vast distances, but that need to have a receiver on the other side. The receivers in question are not trivial to build, and require a fairly advanced civilization to build them.
So, the species in question sends out probes to planets likely to have or develop life. Each probe contains a factory. Once it arrives on a planet, it waits until it detects a civilization has started developing around it. The vast majority of the probes just sit there waiting, until even their rugged systems wear out completely, but a couple of them have managed to find life.
Once a civilization develops, the probe begins manufacturing units that resemble the natives, and using them to infiltrate. It will also start manipulating radio and other wireless broadcasts to further its agenda.
So, in this game, one of these probes has been waiting on Earth for a long time, and awakened with the discovery of radio. It has since then been subtly guiding humanity into building the gates.
Perhaps the probe works differently, seeding planets with the potential for life, hoping that they will develop the gates? This would work for a more Lovercraftian feel, with humanity just being a construction system.
Also, there is no reason why the aliens would have to be invaders, perhaps this is the only way that they can establish contact?
For a twist, in a different setting (perhaps fantasy), the characters discover a similar conspiracy, with the descendants of modern day humanity on the other side of it?