You're correct that you can connect to the internet wirelessly and download music to the iPod Touch from the iTunes WiFi Music Store. To the best of my knowledge, however, you need to sync the iPod Touch to a computer to sync the iTunes Music Store account. It is not possible to buy songs using iTunes gift cards from the WiFi Music Store, these can only be used via the regular store in iTunes. Also of note is that the WiFi Music Store sells only music - your son couldn't download videos, audiobooks or podcasts via his iPod Touch, these are only available via iTunes.
Hope this is of some help...