I also work in an area without access to my phone and without the ability to down load the app. Web messaging stopped working 2-3 weeks ago. I contacted Verizon support last week and they indicated no one else had reported an issue. I see several people have reported the same issue.
I am a PC user.
I cleared cookies and history.
I am using IE 10.
Reporting here - maybe Verizon will get enough with the same issue to resolve the issue.