Hello Kim,
I think the main source of miscommunication here is how you are interpreting "wireless" and "wi-fi". I say that because you keep putting Verizon and wi-fi together in your responses, and they have nothing to do with each other.
I'm guessing this stems from from the confusing use of terms by the industry itself. In this case, Verizon Wireless. Wireless in this context has zero to do with Wi-Fi, or wireless as others are trying to get you to clarify. All Verizon means by that marketing name is that the phone capability of your iPad (and any cell phone) has no wires connecting it to a wall as a standard phone handset would.
You need to disregard anything to do with Verizon for this discussion as they are not in any way involved in the Wi-Fi abilities of your iPad.
Your iPad does two things:
1) It has a Verizon 3G data feature. All association with Verizon ends here.
2) It has a second and completely separate hardware function. Wi-Fi. This part of the device allows you to connect to the Internet.
What others have been trying to get across is that the Wi-Fi part of your device is what you need to use in order to download any updates for your iPad, as it requires an Internet connection to do so. Not a phone connection. It's the same part of the hardware that allows you to use your web browser anytime you are near an Internet Wi-Fi signal.
Whenever you are using the Internet on your iPad, you are using the Wi-Fi, or wireless, part of your device. All iPads have this function, whether you have a separate carrier such as Verizon / CenturyLink / T-Mobile on it or not.
So! What everyone means is if you have no Internet access at any given moment, it means you aren't near the router in your home the iPad communicates with normally, and you haven't gone into the Settings to connect to a different nearby Wi-Fi signal so you can access the Internet when you aren't at home. Again, this has nothing to do with Verizon. The phone part will still work as long as you're within range of any Verizon cell tower even when the Internet doesn't work in the same location since they are not at all related.
Does this help make sense of the issue?