For university awards (e.g. entrance scholarships, merit scholarships): the school/program will/should designate someone in administration to verify. My program identified an administrative assistant that we could use as a verifier. If you program/school does not designate someone, try emailing the awards department at your university or your faculty office (if you come from a smaller program). These awards usually show up on your transcript and are easy to verify for admin.
For research awards: I won a few poster presentation awards and had trouble tracking down the conference organizers. In that case, I used the PI who I worked with on the project (who was also aware of the award).
For leadership/community awards: Contact the community organization to request a verifier.
Email well in-advance so you can confirm someone will be able to verify. I would strongly discourage against using yourself.