Check your mobile data settings and look for "Access Point Names" (or APNs). These are collections of settings that control mobile data access, and if there is anything wrong with these you will lose data access. Since many service providers use different APNs for general data and MMS it's possible for mobile data generally to be working fine and just MMS to not work if the MMS APN is messed-up or not selected.

You should be able to find the correct APN settings for your provider from their website or a quick web search.

I don't guarantee that's the problem, but it's probably the commonest reason for MMS to fail.

It's also sometimes worth checking with your provider that there isn't a problem at their end or in your account settings.

 

I'm not American, so have no idea whether VZW should have just a single APN or a second one for MMS. If it should have 2 and you only see 1 that would be the problem.

I would check their support site, unless a Verizon customer here can provide the answer.