Fallout 76 doesn't have a release date yet—that will presumably be revealed at E3—but it could arrive sooner than anyone expects. The Amazon preorder page currently lists the launch date as December 31, 2019, but as Massively Overpowered observed, when it first went up it carried a "get it by" date of July 31 2018.
That could be an error, or it could be a placeholder that somebody decided should be further back in the calendar. But Bethesda has a reputation for leaving short windows between announcements and releases: Both Wolfenstein 2: The New Colossus and The Evil Within 2 were announced in June 2017 and came out in October of that year, for instance.
E3 to end-of-July is a much tighter window, but Fallout Shelter was announced and released the same day in 2015, so it could happen. (That's a bit of a cheat, as Fallout Shelter is a mobile game, but since we're not sure exactly what kind of game Fallout 76 will be yet, I'll allow it.)
The secret will hopefully be revealed next weekend: Bethesda's E3 press conference will take place at 6:30 pm PT/9:30 pm ET on June 10. In the meantime, here's everything we know about Fallout 76 so far.