According to the mass effect lore, the mass accelerators require careful calculations to end up properly at the far end.
So what you do, is you calculate your coordinates HORRIBLY. Like, TERRIBLY, TERRIBLY BADLY. Mess them up in every way possible. Drop out of the beam halfway through and a hundred light years to the side.
Suddenly, you're in the middle of deep space with no ion trail or any other way of being tracked. Then you just use a solar sail to cruise to a likely garden world(so you don't have any emissions) and bury your ship under a couple hundred tons of rock and under an ocean. No way emissions will get out through that.
Have your people go into sequential hibernations, with 99% of them asleep and 1% awake for 5 year periods. After all of them have been awake for five years, the reapers will be done with their harvest and will have returned to dark space. Oh, and did I mention, you've been mining using conventional equipment for all that time? Now you have a massive accumulation of resources, absolutely no competition, and the entire galaxy ahead of you for 50000 years. Of course, you have to lay low and avoid watchers like Sovereign, but a single ship simply cannot expect to locate a civilization that's focusing on being hidden, especially if you're multiple years away from any mass effect gate.
So you spend the next 50000 years building enough weapons to cover the surface of every planet you colonize. You enact laws that make every building have to have a centralized mass accelerator. You excavate hundred mile long tunnels through moons from all directions and build mass accelerators through them pointing in every direction, so you can fire without the problems of having to turn.
Finally, you fail to be quiet. The reapers show up with a light expeditionary force.
They are instantly annihilated by a million mass accelerator projectiles moving at 99.9% the speed of light. They arrive only moments after their light does, giving them no chance to react, and the energy released would be sufficient to reduce the reapers to their constituent atoms. No warning would escape.
Several years later, a larger force arrives. It contains several dozen Harbinger-class Reapers, as well as hundreds of smaller class reapers and millions of ground assault troops. Their cargo is irrelevant; there are thousands of mass accelerators pointed at them, capable of unleashing a destructive force powerful enough to match the energy output of the sun. At this point, it would be no surprise if it IS the energy output of the sun; 50000 years is plenty of time to construct a Dyson Sphere. The entirety of the Reaper Forces could attack at once and it would make no difference; the collateral damage via energy released from the destruction of the first ship would be so great that it would almost certainly cause the destruction of the others.
Of course, this is all assuming the reapers even find you at all. Given the 50000 year window, you could easily build a complete dyson sphere and start regular mass-energy conversion to build any type of matter you want. By the time they return from their sleep, what once was a hot sun is now a cold grey sphere hurtling through interstellar space. Inside has enough surface area to support every sapient creature in the galaxy for years without ever seeing another living soul.
Next cycle comes, you send out scout ships to collect all the species that want to join your group. Many join you, and after a few cycles pass, you can show conclusive proof that staying on their homeworlds will result in their destruction. For several cycles the Reapers return, only to be confused by a galaxy already long empty. Finally they find you; through thousands of years of enforced peace, your people have slowly integrated themselves with each other, striving for the goal the Reapers could never achieve; true peace. Eventually even minds are linked together, forming one great collective consciousness, all while you modify your fleshy bodies with implants. Wishing for mobility, you've created new ships, in the model of your home sphere as well as in ergonomically pleasing cubes. Finally, you rename yourself, choosing a name more fitting to the amalgamation of races, cultures, and thoughts that is your people. Soon the time comes; time for the lost sheep to return to the fold. Time to reclaim the Reapers. And at last, watching as their ships are annihilated in seconds while your uncounted trillions of minds compensate for their attacks in instants, the Reapers ask you what you are. And as one, you reply;
"We are the Borg. Lower your shields and surrender your ships. We will add your biological and techniological distinctiveness to our own. Your culture will adapt to serve us. You will be assimilated. Resistance is Futile."