The chess.com business model depends on locking people into using their site to play through in order to convince those users to purchase subscriptions to the site (just as is the case with Chessbase's Playchess site, the ICC, FIDE Online Arena and several others). Opening up access to their API (they do have one for their smartphone apps) in the way you suggest could be readily used to break through those commercial barriers. Which is a round about way of saying, no, there's not much chance of using any other program with their site.
Is there any way to tweak it so that I can connect to chess.com's server?
Of course. Just use Fiddler to monitor the HTTP traffic to chess.com from your desktop/mobile phone/tablet to see how it can be done. It is technically possible.
I don't think there currently is a way judging from the discussion in the following link:
http://www.chess.com/forum/view/community/chesscom-desktop-client?page=1
(Or using any other interface, for that matter.)
Without intercepting and analysing the network traffic between you and the site - basically reverse-engineering the API, then writing your own client - you can only use a web browser to play at their site.
Note that you can't even use any web browser, they suggest only G@@gle Chrome or Firefox and even these only in certain versions. For instance, you can't play using Waterfox (a Firefox fork), because it is blocked. I suppose they check the "user agent" string (for instance: see you user agent).
If you want to use any other browser (version) you might get away with modifying this string. In Firefox this is done editing the configuration: enter about:config into the address field and add a new entry:
general.useragent.override
of type "string". Here is a list of the user agent strings of various browsers.
Be careful, though: first, different browsers have different abilities and some websites tailor their sites contents (spcifically scripts) according to this value, so it might break some sites. Second, there are other ways of determining the type of browser you are using too, so overriding the user agent might not always help.
