Yes, there are several methods of viewing text messages online. Some services offer text message forwarding to an email or messenger account, while others require logging in to an online cellular carrier account.
Google Voice is an example of a service that can forward text messages to an email account. This requires a Google Voice account, and can be implemented via the text settings menu. Yahoo Messenger allows texts to be sent via a phone app that can be viewed online. Both Verizon and AT&T users have limited access to text messages online by signing into their cellular accounts. Verizon allows users to view responses to past text messages, and replies can be sent online without a phone. AT&T accounts will only display information about a text, such as date and time, but will not show the contents of the text messages. This information is accurate as of 2014.