You can't have disabled the iTunes Store and still get a free app, so I'm not quite sure what you mean. In regards to the charge, though, you probably inadvertently made an in-app purchase. Many apps, particularly games, are free for the base game but charge for additional features such as levels, "coins" and other such add-ons. First, to prevent this from happening again, turn off "In App purchases" in the Restrictions settings on your iPad. For more information, see:
http://support.apple.com/kb/HT4213
As to a refund, that's not automatic since the terms of sale for the iTunes Store state that all sales are final. You can contact the iTunes Store, explain the reason for your request, and ask, though:
http://www.apple.com/support/itunes/contact.html
They're usually pretty lenient in the case of inadvertent purchases, but there are no guarantees. Definitely contact them if you are certain you could not have made any in-app purchases.
Good luck.