is it possible to have the customer enter a name as an attribute?

I do the website for a local yoga studio. For the holidays, they'd like to offer some specials such as a
yoga mat and 3 months membership, or a tank top and 3 months membership. The customers can go online and purchase them, but the yoga studio isn't shipping anything, everything will be picked up at the studio.

Pretty much anyone who will buy anything will be going to the studio anyway, so its a reasonable approach.

I'd like to make it so that when someone goes to the online store to purchase something, there would be a checkbox for whether or not the item will be a gift, and if that box is checked, then the customer will be given a textbox to fill in the name of the person receiving the gift. That way when the giftee goes to the studio to claim it, the studio will be ready for them.

Is this possible?

It would take custom coding to make the textbox only appear when the checkbox is checked, but you can just as well have language like "if you want to give this as a gift, enter the name of the recipient in the textbox". As long as you don't make that textbox mandatory, it will be ignored if not filled in, and saved as an attribute if it is filled in.