Where is this Web Messaging app through? Are you referring to the Integrated Messaging offered by Verizon Wireless? If so, the messages are functioning as designed. The messages sent as iMessages are sent through the data connection and Apple's servers. They don't go through the same VZW system as regular text messages that would show up in the Integrated Messaging.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.