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Your predicament in which a provider requires a browser other than IE sounds like bad planning. Why anyone would require you to use something other than the most popular browser will come back to bite them.

As for the Help Desk Ticketing situation:
I think you understand why a ticketing system exists so I won't belabor the point. What I will say is that the Help Desk Department can and should make help desk requests easy for the user by enabling several ways to reach the Help Desk.
Our users can enter the ticket directly, email a narrative to the help desk email account, vist the department, email, call, or bump into any one of our IT staff who will help them and log the ticket into the help desk system for them., If no one picks up their phone in the help desk hotline hunt group, their voicemail is automatically entered into the help desk system as a wav file attachment. We have a written triage policy that lets the user know how soon someone will get in touch with them and how soon the problem should take to resolve. My staff will visit the user to resolve their problems rather than do "walk throughs" over the phone. It's not because we don't have tools to fix issues remotely. It is because making my staff to be visible makes the user feel appreciated.
We don't have a pretty boat to transport people to the Help Desk Department but that's only because senior management removed it from the IT budget.