I believe you have to buy the app through iPhone, Roku, etc, but after you purchase through that, you are able to watch it on a desktop through hbonow.com. I agree that this method is really dumb though. I was trying to purchase it last night for the GoT premiere, and it took me almost 20 minutes just to figure out they didn't have an option to purchase directly from my web browser. Not only that, but I got stuck in the "Renew your Subscription" loop, and still can't even watch anything, so until they get that fixed, I'll be pirating the episodes.
Just doing a wiki walk, it looks like you can get what you want through "HBO Now". Says it's video on demand, works on tablets and through web browsers. Official website here.
As for the rest... just tell them directly what you want and ask about pricing. There might be a chance of paying more, but I doubt it (though you may have a termination fee).
The way it will work is if you have an IOS device you can get HBO Now, which is $15 a month.
Edit: HBO Now will be exclusive to IOS devices for the first 3 months AKA the Game of Thrones season
You can stream the entire Game of Thrones Season 7 on your PC for $14.99 on http://www.hbonow.com
Your first month will be free, then you pay $14.99 for your second month. Season 7 is 7 episodes so it should finish during your second month. Then you cancel your subscription at the end of the second month. You have to register for HBO Now via the iOS or Android app.