Here are a few:
webaward.org — Longest running(17 years) annual website award competition dedicated to naming the best Web sites in 96 industries.
cssdesignawards.com — International web design and development awards that honor and showcases creative work on the web. They give out daily, monthly and yearly awards are presented for outstanding creativity and technical proficiency.
thefwa.com — They showcase the most progressive and forward thinking projects, created by individuals; agencies; and brands.
FWA gives out awards to Websites, Apps, VR projects and other creative works that push the boundaries
csswinner.com — Global platform awarding and showcasing the best websites and promoting innovative web designers, developers and agencies.