If you are trying to connect the Time Capsule using wireless only.....(I do not recommend this)....you must know the exact type of wireless security settings that the Verizon wireless network is using.
Then, those settings must be provided to the Time Capsule during setup. If the settings are correct, you will see a green light on the Time Capsule after it updates.
Do you know the exact type of wireless security that the Verizon router is using? If not, can you check with Verizon on this?
We can go on for hours trying to find a compatible setting between two routers....but this is one of the reasons why I do not recommend this type of setup for the Time Capsule for most users.
Want to make sure that you understand that the Ethernet ports are not enabled on the Time Capsule in this type of setup. So, no option to back up large files using Ethernet...which will be 3-5 times faster or more compared to wireless.
WPA has different flavors and variations. Might be WPA-PSK-TKIP or WPA2-PSK-AES settings on Verizon...I do not know.
What choices do you see on the wireless settings page for the router?
Are you trying to configure the Time Capsule to "Join a wireless network" or maybe "Extend a wireless network"? ("Extend" will not work with the Verizon router)