I migrated to AOL from Thunderbird.

Thunderbird is email client software running on your computer. Thunderbird is not an email provider. So there's no way to migrate from Thunderbird to AOL.

I have clicked on the "GET MESSAGES" tab, thinking that at least a sign in window or configuration window would pop up, but NOTHING happens.

What is the account type of your AOL account - POP or IMAP?

Are your messages still available on the server? You can check by logging in to your AOL account with a web browser. https://mail.aol.com/

I think I would rather have an IMAP account. My understanding is that with this type, if you delete an email on one device, it will delete it across all devices that share your email, whereas a POP will not.

My messages are still available on AOL and are current.

My biggest problem is that I have several addresses on Thunderbird that I was not able to export to AOL, including one group email. I need that group email badly.

My understanding is that with this type, if you delete an email on one device, it will delete it across all devices that share your email, whereas a POP will not.

That's correct. However, the question was about the current account type. So I'd assume it's POP.

You can follow these instructions to set up your AOL account as IMAP. http://kb.mozillazine.org/Convert_a_POP_account_to_a_IMAP_account

Once the account is set up as IMAP you can copy any messages from other existing accounts to the AOL IMAP server.

Also make sure to check with Verizon/AOL about any changes needed to keep your Verizon email address working.