This is only about the previews.
iTunes does embed your Apple ID (you need to sign up and get one to redeem your gift card, though you do not need to enter any credit card info or the like) and purchase date as well as time as metadata in the file, but that information is easily removed. There are no watermarks.
Is there some sort of iTunes kiosk, a publicly accessible device with iTunes installed where a customer could purchase songs, download them to removable media and then move the songs from the media to whereever they need to have them?
Or maybe you can use a friend's device with iTunes installed?
Yes, you need iTunes to buy – but once you have the music, you do not need it anymore. They are just standard audio files that work with any (somewhat current) player.