That would certainly depend on how you manage your messages.
IF you are subscribed to Verizon Messages and have it installed on your iPhone, you can view them online.
If you are not subscribed and you want to do so, I don't believe any messages already sent/received would be included in the online stream. For ALL messages to be included in your online stream, you would have to disable iMessage as any messages sent/received from Apple's iMessage servers would not be available. Once iMessage is disabled, all of your messages would be sent/received as SMS/MMS and would be available.
I don't know if Apple has this capability with their iCloud accounts.
rcschnoor is on point with Verizon Messages. Once you register your line via downloading the app in the App Store you can simply go to vtext.com and login. This doesn't include iMessages, which you can toggle off in Settings.
If you have a Mac computer you can register Messages to accept your text messages from your iPhone. I use Messages on my Mac to send/receive texts, pics, iMessages, etc.
If you start using Verizon Messages you can download the client to your PC and text from there as well, which is the same as how Messages works on the iMac. Take Texting to the Next Level | Verizon Wireless