I don't have passport and no driving license. I was wondering if I can use Aadhaar card as ID proof for opening account on online bookies?

They should accept it as it's national ID card. You can upload or email them and if they don't accept then explain that it's now the national ID and point to the website. You can also contact them via online chat if it's available. 

Well they did in my case .Not on Betfair but on Skrill, since you need two proofs. One for the identity and one of address proof.