Internally, we use Bugzilla. In "enterprise" contexts, the "one ITIL to rule them all" was like Maximo at the bank, and Project Open in one or two places. Project open would be my short-list.
I had this exact same question about a year ago, and I suspect that you will find the same thing that I did when I asked /r/sysadmin - nothing. All suggestions here are either SaaS, paid-for software, hosted, or cannot support more than one user. I spent a week of my life demo-ing different products and was having a heck of a time.
If you are looking for something that is fully-featured, open-source, web-based, and internally hosted, then I have just the thing for you. It's called ]Project Open[ - http://www.project-open.com/
Warning: It is clunky to figure out when you first deploy it. If you give it some time and begin customizing the modules (e.g. time-tracking, user administration) then you will quickly figure it out. It comes as a fully-featured project management platform, but you will want to pare it down to just user admin, time tracking, projects, and time reporting.
Ah, integration into that level would be quite complex, running in the $25,000 - $75,000 range to build a custom ERP / CRM suite on an open source platform (Feng Office or ProjectOpen would be a good start), or $150,000+ on something more commercial like Microsoft Dynamics.
If your staff is already used to FieldOne, it may make more seance to use their open database integration and pay a developer to tie your FieldOne client and user databases into one of the prior mentioned help-desk softwares.