Can my iPhone 13 mini use Verizon network?

If your iPhone 13 Mini is SIM locked to your current carrier, you must have your iPhone SIM unlocked before you can use a new carrier. Contact your current carrier to get your iPhone SIM-unlocked.

You can find more useful information here:

How to unlock your iPhone for use with a different carrier - Apple Support

Who is your current carrier? If you bought the phone through them, and phone is not yet paid for, the carrier is not likely going to agree to remove the lock on the phone.......which must be done if you want to switch your service to another carrier.

Yes. Verizon Network should work with your iPhone.