If you're using Verizon Messages, the content is available via your on-line My Verizon account. Once logged in, hover over My Plan and Services at the top, then click Send a Message.
If you aren't using Verizon Messages, only a text log is available (and only if you're logged in as Account Owner/Manager); to view the log for the current billing period, click the View Usage Details button. You also have access to the previous 90 days' worth of text logs via your My Verizon account. Hover over My Billing & Usage at the top and then click View Bill. Use the drop-down to select the billing period, then click the Calls & Messages tab, then click Messaging.