No. You will need an Apple ID to download apps from the App Store. When you set up your Apple ID you can set the default payment type to “none.” In this way you can have an Apple ID without having a credit card. You will not need a payment type listed to download free apps. To purchase apps without a credit card, you can use an iTunes Gift Card. These cards contain a code that can be redeemed using the App Store or iTunes Apps. The U of M Bookstore sells iTunes gift cards.

See also:

Do I need an Apple ID and how do I set it up?

How do I use the App Store?

How do I redeem an iTunes Gift Code?