My first guess is that you're (sensibly) having your boy run as a non-administrative user. Trouble is with Windows that this really doesn't work, since most software installation requires admin privileges, so folks usually just go ahead and run as admin to avoid the headaches.
The most sensible solution is to use a true multi-user operating environment and not one which has some semblance of the design stuck on with bubble gum after the fact.
Either Mac OS X or Linux fit the first description and you know what fits the latter.
The route the data takes from the web site to your son's computer really shouldn't be a factor.
I'd be a little concerned about having a 6-year-old browsing around the 'net unsupervised, especially on a high-speed connection (and most especially on a Windows computer). In order to not have to feel like I'd need to watch over his shoulder, I'd set up a headless Linux or FreeBSD box on the network through which all traffic must flow, and on which a filtering proxy is running; one such as I use here to re-write incoming web pages to omit most ads, most intrusive data-gathering stuff, and most annoying stuff such as unrequested popup windows.
