The Storm Team are an elite EDF unit. The name is applied to the 1-4 player characters in Earth Defense Force 2025 and Earth Defense Force 4.1: The Shadow of New Despair.
"Storm Teams" are mentioned in the mission briefings of Earth Defense Force 2017 and appear to be an elite EDF group, the equivalent of the American Delta Force or British SAS, as opposed to the Rangers who are standard Army infantry. Storm Teams other than Storm 1 have multiple members and Storm 1 is called a Storm Team in Tsugawa, implying his team is just chronically understaffed. Some levels also refer to Storm 1 as a "they," confusingly.
By the time of 2025 it seems there is only one team that goes by the name. They fight alongside the Japanese EDF forces under the somewhat erratic command of HQ, a figure at the Strategic Command Headquarters, which seems to be a local EDF command rather than the central one.
In the story introduction in EDF 2025's EU-only manual, it is mentioned that the hero of the last war, Storm 1, disappeared, and during the story EDF soldiers express doubts that the stories they have been told about one man taking down a Quadruped Fortress and the Mothership are true, with one saying they have just been blown out of proportion. However, it is later revealed that "that man survived," and one soldier comments that the hero is said to have joined the Storm Team, with the implication being that Storm 1 is whichever character is being played as. Presumably if the character being played as is a Wing Diver, it was actually the Pale Wing who took down the Mothership in 2017.
More confusingly, during the final battle the leader of Omega Team says he fought alongside the entire Storm Team in the last battle of the 2017 war. This could be seen as implying that the stories (and by extension EDF 2017 itself) actually are exaggerated and Storm 1 was always more than one person, though it is not clear if this is the intent.
In Earth Defense Force 5, which is set in a different continuity from 2017 and 2025, the title appears late in the game as the Rangers of Alpha Team, the Fencers of Grim Reaper Team and the Wing Divers of Spriggan Team are assaulting Base 228 to re-take Gigantic Unloader Barga. Seemingly just to stop Spriggan and Grim Reaper arguing, HQ assigns them new codenames: Spriggan Storm 4, Grim Reaper Storm 3, the Sergeant (though seemingly not the rest of Alpha) Storm 2, and the player character(s) Storm 1.