If a person grew up in the US but has no ID or birth certificate and cannot prove their residency, are they legally a citizen?

Growing up does not equal being a natural citizen (someone who is born in US) or being a naturalized citizen (someone who applies for citizenship and obtains citizenship based on being a permanent resident for a period of time). 

You need to first figure out -based on your comment that "a person grew up in the US" -whether you were a green card holder/permanent resident or not.