Hey Chris!
Absolutely this is possible, but the information would need to be added on to the product prior to it being added to the cart. For this, I'd recommend using BoldApps' awesome Product Options app.
http://apps.shopify.com/product-options
The main advantage that this app has is that options, or variations on a product, can have an affect on the variant's price, and so if you wanted to charge more for this customization, this app would do quite nicely.
Let me know what you think!
Matthew Buttler
Guru
Shopify Inc.