Published instances of phrases of the type "X High School varsity football team" in Google Books search results show considerable variation in initial capitalization. The vast majority capitalize "X High School" (presumably because the authors recognize that they are dealing with a proper name). As deadrat's answer implies, most then split between all-lowercase for "varsity football team" or all-initial-caps for "Varsity Football Team." But one Google Books result opts for "Window Rock High School Varsity Football team" and another for "Vallejo High School's Varsity football team."
I wouldn't capitalize "varsity football team." The team is certainly an offshoot of an entity that has a proper name—say, Francis X. Bushman High School. But if we accept "Francis X. Bushman High School Varsity Football Team," why not "Francis X. Bushman High School Left Field Fence" or "Francis X. Bushman High School Volleyball Net" or "Francis X. Bushman High School Recycling Bins Outside the Lunch Room"? Those are identifiable entities associated with the school. too.
Incidentally, the word varsity began (around 1646) as an abbreviation/altered spelling of university, and some examples from the early 1900s retain the apostrophe before the v: 'varsity.
Ultimately, capitalizing Varsity and/or Football and/or Team is a style issue that every person who is free to act on personal preference gets to decide without reference to what others do. But at the level of proper nouns, I see very little reason to treat "varsity football team" as anything more than a string of common nouns, whether or not it follows in the shadow of a school's proper name.