I would suggest using the python trac package, to quote:
Trac is an enhanced wiki and issue tracking system for software
development projects. Trac uses a minimalistic approach to web-based
software project management. Our mission is to help developers write
great software while staying out of the way. Trac should impose as
little as possible on a team's established development process and
It provides an interface to Subversion and Git (or other version
control systems), an integrated Wiki and convenient reporting
Trac allows wiki markup in issue descriptions and commit messages,
creating links and seamless references between bugs, tasks,
changesets, files and wiki pages. A timeline shows all current and
past project events in order, making the acquisition of an overview of
the project and tracking progress very easy. The roadmap shows the
road ahead, listing the upcoming milestones.
Full installation instructions are here.
If you take advantage of the VCS integration then you can tell which changes to the code, by whom, resolved each issue or implemented each new feature.
Trac is free, gratis and open source.
The main user interface is via web pages so if you have tablets or phones that can connect to the companies intranet you will be able to edit the details of tickets, check for additional open tickets near your current location, etc., via them.
Any such system is only as good as the data that is given to it - this needs discipline and either a willingness to share information or a set of rules to make sure that the information is shared. Using a wiki, (which trac provides), is one way of enabling the sharing of information but if nobody puts the information into it it will be useless while building the use of tickets, (which trac also gives you), into your workflow and ensuring that they are filled in properly, possibly by most people only being able to mark tickets as "ready to close" and checking the details provided before actually closing the tickets. While this requires discipline and people will moan about the additional overhead but the benefits of taking a more professional approach will be great as you will build up an institutional knowledge base, as opposed to individuals building a personal knowledge base, that will be more and more valuable over time.
It will also allow the generation of actual statistics that will be valuable, e.g. being able to show that you are getting 25% more incidents per month is a lot more likely to get you the funding for additional staff than simply saying "we are busier than ever".