how can i print out old text messages?
The availability of options depends on the phone model and operating system. What is yours?
As Ann said, it does depend on your phone. One third party app that works on many phones (especially older models, but not all phones) is BitPim ( http://www.bitpim.org/ ) but it can potentially harm/brick your phone if you don't know what you are doing - use it with caution. I have used it successfully to get texts and photos off an old flip phone that was deactivated and had no sd card.
Newer smartphones can download apps that will give you the ability to print out texts.
lg 4glte
Is there any way to print texts messages from your Verizon account
No, you can only access your call log via your on-line MyVerizon account. You could forward the texts to your e-mail and print them from there. There also are apps in the Play Store that will back up your texts and forward them to your e-mail.