Many cities and counties now use software to manage incidents from 911, police, and fire departments. These are many industry buzzwords for these, including “emergency management systems” and “computer-aided dispatch”. Local police and firefighters will be very familiar with their local version.
Many of these systems require dispatchers or responders to code each incident by type. When the system has a code specifically for homeless camp fires, as distinct from other kinds of fires, that can create useful data. You might be able to get that data from city or county through a public-records request.
For example, here's the online record for a records request of homeless camp fire incident data in Oakland, California. That data could then be analyzed to put numbers on the scale of problem, map out “hotspots”, and confirm intuitions about when during the year more fires happen.
If your city or county doesn’t yet have a code for homeless camp fires, you might convince them to start.