We have a customer that is disputing 41 ITunes transactions totaling a little over $1,200. We feel it is someone in the family doing the transactions. Can we ask the customer to bring in her purchase history for the last two months before we move forward with the investigation/reimbursement?

You can't *make* the customer provide you that information as a condition of your investigation, but you can certainly ask them for it.

Hiker is correct. Here's part of a position paper we are currently writing on this topic:

A bank only needs to receive the following for Regulation E requirements to apply.
§1005.11(b)(1) states, a financial institution shall comply with the requirements of this section with respect to any oral or written notice of error from the consumer that:
(i) Is received by the institution…
(ii) Enables the institution to identify the consumer… and;
(iii) Indicates why the consumer believes an error exists…

Bank’s may certainly request a customer's cooperation in the investigation anything beyond the above can't be a condition to starting an investigation or granting provisional credit. Banks can turn the transaction over to the police for investigation and they may require a report to be filed, but again, the bank cannot tell the customer they have to file a police report to make a claim with the bank.

Sorry for being late with my input on this. I have had very good results by having the customer call iTunes and report the fraud. They refund within days, usually within the 10 day Reg E timeframe for provisional credit. I still do my disclosures, letters, etc... under Reg E just to be prepared in case it doesn't work out.
I found that to be easier and cheaper than eating charges that are, for the most part, below the MasterCard dispute limit.
The same is true of most big companies, such as Google Wallet, Steam Games, etc.

What Cecile states may work, but you need to be careful that you cannot require the customer to contact the merchant. Reg E basically says "Your card, your customer, your problem."