This might be a dumb question, but I'm assuming Mozilla is set as your default browser for the whole computer? Sometimes programs are 'locked' into using only one browser (usually IE) and can't be changed, I've seen it with other programs on my PC.
Other thoughts:
how up-to-date is your copy of Explorer?
maybe your Mozilla has an add-on that you've not installed in Explorer because you don't use it as much?
What version of Windows are you using?
I've just successfully downloaded an item from the store using Vista with an completely updated IE so it should be possible on your machine too.