Bryce seems really hyped up that he "called Ohio last week", so it seems to be set in the late 20th century (the Voyager took off in 1977, which is referenced at the beginning of the film). This is reinforced by one of the computer files having ".mp1" as its extension, which was used mostly in the '90s (being published in '93).
There is an ofrenda seen, so clearly it is set in late October (when Elio knocks things off the ofrenda, his aunt doesn't react much more than if it was any other masterpiece, so clearly Día de los Muertos hasn't begun yet).
Since most of the movie takes place in space, there isn't much to go off, so I'm happy placing it in an October, during the '90s.