In my country of Norway online gambling is legal, but the banks are not permitted to transfer money to online gambling companies. So all transfers of money to online gambling sites are done thru third party. is it possible to set up a system using bitcoins to allow people to transfer money to and from a online gambling site?
I'm not a gambler, but it's probably doable:
It depends what sort of payments your chosen gambling site accepts. If they accept bitcoins, then simply buy some BTC at an exchange funded using your bank account, then withdraw your new BTC from the exchange directly into your gambling account.
If you chosen gambling site doesn't accept BTC, you could always still transfer it via and exchange, although this seems a bit silly as there are probably cheaper services to send the money via.
With this method, you'll obviously incur transaction fees at the exchange, as well as deposit/withdrawal fees, and possibly currency conversion fees.
Another alternative that wouldn't use Bitcoin would be to send the money via TransferWise. TransferWise recently stopped handling Bitcoin related transactions, they also may not be happy about sending money to gambling websites, although I'm not sure.
tl;dr Yes - convert it to Bitcoin and back into fiat if that's all your gambling site accepts. Could prove costly though.