Different carrier

Yes. You'll need to cancel your plan with Verizon, and then either setup a plan directly on the iPad with another carrier, or go to another carrier and request a sim with a plan from them.

Details here:Set up cellular data service on your Wi-Fi + Cellular model iPad - Apple Support

What says its not available?

If the carrier you wish to use, is not an option within the Carrier settings on the iPad, you will need to go directly to one of the carriers store and request a sim card from them.

In the U.S, at least: AT&T, T-Mobile and Sprint are supported for direct setup on the iPad. Other's may require you go to a carrier store and get a sim card from them.

More details here: iPad - Cellular - Apple

Not sure where you are from.

You'll have to look at other carriers.

AT & T, Sprint, T-Mobile, Family Mobile ( through Walmart ), Cricket Wireless, Boost Mobile...

I am not familiar with every single wireless carrier and if all have data plans, or not.

You'll have to do your own Web search.