I wouldn't rely on the "word" of the copyright holder; get an actual legal contract. But, yes, if someone allows you to build a shell around their product, and you have it on paper, than you should be fine. IANAL, so get one.
Setting up a proper, secure (money is involved, which will attract criminals), web-based application, and its infrastructure is not easy when you have never done any coding before.
Also, building a shell around an existing, suitable, cross-platform game is unlikely to be easy. You'd have to find the right game, with open licensing or with a kind-hearted copyright holder, and with a programming interface (API) that allows you work with the game from your wrapper. Not an easy hunt!
I still think your best bet right now is to go to YoYo Games. Or, Kongregate. Also, you may want to consider finding a Flash coder in Bangladesh to put this together for you. (Seriously. If you are cash-strapped, this would bootstrap it. And, yes, there are a slew of ethical and management issues on outsourcing...)