For the sake of ease, I'm going to assume that the writers are going to simply put it down to "The Reapers never left the Milky Way because their programming only dealt with the Milky Way." Which, to me is a perfectly logical point. Inter-galactic travel is tricky at best, and a well-established group of people would only want to leave because they had to, such as in the case of Ryder and Co.
This is what I think was/is the reasoning for why Ryder and Co. leave the Milky Way: Before the events of ME3, a bunch of people belonging to every race and species (or at least the ones well represented) gather together - they have one common thread of believing that the Reapers will invade and will kill everything. So they pool their resources and create the Andromeda Project, or whatever name they're going to have for it in the next game. Maybe they'd have started this project after the initial attack on the Citadel, maybe they'd have started it beforehand after discovering the Reapers some other way, but either way, the Project is to ensure the survival of their many species and cultures, and for the sake of security, only a few people know about it. So before the events of ME3, they take off into the depths of Andromeda without looking back because, hey, it would be safe for them to assume that everything back home is going to be very shortly f*cked up by a bunch of machines, and everyone else is trusting in a person (Shepard) who may or may not (but probably won't) be able to save them. It would make sense for them to try to put their origins behind them, because as far as they know life as they knew it back home is gone, and in some ways they're right, given the different kinds of endings there are.
But my point there was that this group (if my belief regarding the timing and reasonings for their leaving are correct) are only leaving the Milky Way and trying to start a new home is because they feel that they absolutely have to in order to survive. If they thought they could stay in the Milky Way and not get turned into a bunch of gooey paste, they probably would have. They left only under extreme duress so that their people wouldn't go extinct like so many others before them.
Now, let's look at the Leviathans and the Reapers. The Leviathans had it good. Real good. They ran the known galaxy and had everything they needed. They had no outside threats or any species that could compare. Would some of them have tried to explore another galaxy? Possibly, but why would they, outside of pure curiosity? They had everything they needed. Who knows if they could even survive in another galaxy. It would be a risky venture costing lots of resources with questionable payoffs. I just don't see them bothering with exploration outside of their own galaxy (which hasn't been explored completely even up to the point of the first game if I'm remembering correctly), especially given their attitudes towards being the greatest thing to happen ever.
And the Reapers were originally created by the Leviathans. If you wanted to get really specific, you could say that the Intelligence made the Reapers, but the Leviathans made the Intelligence, so either way they started it. Why would the Leviathans bother with programming their creations with anything other than "preserving life" which to them would have been within that one galaxy? The Reapers were a means to an end- they have spent millions of years harvesting life within the galaxy on that one mandate, so why would they bother exploring other galaxies? Could they? Without a doubt they could, but they don't think like an organic life would, they think within the 'limitations' of their programming, which I think pertains to the Milky Way galaxy, and only that galaxy. If the Leviathans wanted to explore other galaxies, they would have made different machines to explore, or they would have gone themselves. The Reapers were intended for a much more basic purpose that went completely out of control.
Personally, I think it's more likely that we'll encounter other races who also made an exodus from the Milky Way to escape the Reapers, rather than finding Reapers/Leviathans already there. Or maybe we will find evidence of a Leviathan expedition, but I doubt we'll see anything significant from the Reapers in the next series. That chapter is over and done with (thank God).