It's not the "fault" of lego.com, et. al. It's the unfortunate consequence of
1. Microsoft is allowed to bulldoze competing browsers out of the market,
2. Web sites that are trying to make a buck make their applications work with the dominant browser, rather than with standards.
3. Microsoft doesn't care about standards, indeed, deliberately skews them and invents their own proprietary features, so that web sites only work with said dominant browser.
It's just the state of things. Also, unfortunately, an 8-year-old will neither understand nor care. I do hope the plug-ins suggestion below works for you. Other than that, is it possible to run IE under Wine, and will those sites work that way? Seems like a long shot, but I would at least try it.
After that, the only other thing to try is running XP under VMWare. Ugh - more money to spend. It's a trade-off you'll have to evaluate yourself.