Traditionally (in the comics), he had access to lots of equipment at college. He would work late in the lab and build his stuff by tying in a similar project he was working on. On paper, it just looked like he was just working on school projects.
Toby McGuire Spidey didn't have anything fancy that I can recall. His body shot webbing.
Andrew Garfield Spidey, well, I don't recall them making much sense for anything that went on in that series.
And Tom Holland Spidey was wearing underoos until Tony Stark found him.