Can I use a Verizon iPad 3rd Generation WiFi Only?
Yes, you will be able to use the iPad without a Verizon SIM card through your WiFi network. However, it appears that there is a problem where the iPad keeps notifying you that there is no SIM card.

I would advise getting one without cellular data as you don't plan on using it anyway. Those iPads are probably cheaper and will not have the previously described problem.