The church doesn't not have insurance on missionary iPads/tablets. However, in the new android tablet program (my mission had it rolled out to them 9 months ago), missionaries can buy their Samsung S2 tablets with insurance from the place of purchase.
My mission office has a loaner program for missionaries that can't afford, or can't afford to replace, an iPad/S2 tablet. The missionary can borrow a tablet from the mission office.
That being said, you will need to contact the mission office of your Missionary's mission for more details about your options. You likely have more than you think! Please reply if this helps, or if you have any other questions.
-Elder Thomas Parker, Colorado Fort Collins Mission
I'm just a regular missionary. I love it when the original post's author never says "it worked!"