Straight Talk is owned by America Movil, along with Trac Fone and Net10. Unlike most other MVNO's, they have contracts with all four of the Big Four carriers, though any individual handset will ride on only one of the four networks. The specific assortment of handsets they'll offer (either online or at retail) in any given location is a function of the coverage available from the Big Four at that specific location. Hence their website asks you for your zip code before showing you phones.
The retail packages indicate whose network the phone runs on. "CDMA-V" is Verizon, "CDMA-S" is Sprint, "GSM-A" and "GSM-T" are AT&T and T-Mobile, respectively. Pick the right phone, and you will be riding full-time on Verizon's network while paying Straight Talk's cheap monthly fees (and putting up with their shitty customer service).