Aha! Thanks for doing the heavy lifting @Fintroop.
This is indeed from Legion of Superheroes Vol 4 #88. The Legion is time displaced in the past and are staying at STAR Labs in the meantime.
A congressional oversight committee is inspecting their facility as Impulse (who dropped in wanting to join the Legion) helps Brainiac 5 with an experiment, but ends up triggering a core meltdown.
The Legion prevents STAR Labs’ destruction but are evicted from their temporary home, and Impulse is denied membership (does that all sound right?).
So technically, the Legion is right by saying Impulse is the first applicant denied entry, since he failed to get in during the 20th century, and the other try-outs applied in the 31st century.