They do it indirectly, without ever having to visit a polling place or pull a lever. It seems that our Representatives are apportions according to the total population of the country, currently divided by 435, which says that each representative represents about 720,800 people. Note that we're talking people here, NOT citizens. Now in the electoral college, votes are apportioned out one per Senator, or two per state regardless of population, and one per representative. If your state has a bunch of non-citizens in it at the time of the census, you get extra representatives, for example, California gets 5 extra.
I did some looking and some math. 315M people divided by 435 reps is the above mentioned 720,800 people/rep. According to the Kaiser Foundation, some 7% of those are non-citizens or 22.5M. By current estimates, about half of them are illegal. Dividing the 22.5M by 720.8K tells me that non citizens have 30 representatives and 30 electoral votes to their credit. Given their propensity to vote Democratic, that puts the Republicans at a distinct disadvantage right out of the blocks. Toss in a bit of creative gerrymandering and you can use imported non-citizens to push voting citizens out of a close district.
They don't have to vote. They get representation for free.