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A police report is required, but not sure what it covers.  You probably already know this, but you must reside in Belize continuously for one year (with only 14 days out of the country) before applying for permanent residency.

wejust appliedoours in Corozal.... so far it has been 6 weeks and t report i not back yet. They will tell u it taes apprx 4 weeks, that mean up to 8 weeks lol. if applying in Corozal, u need to have a copy of your pastport picture and copy of first entry stamp and lst visa stamp- mut show 1 year in the country. 3 photos (canbe gotten and Dave's photos) go to polics station there is a side  window on the building, go there give the lady all docs, pictures and tell her you need a crim search done for your rest. and than wait
after 4 weeks.. check bck at the same window to see if it is in...if not, check bck weekly thereafter until u get it.

Got ours around 10 weeks after requesting it (BZ$12 per person). We were initially told it would be 4 weeks.
Got ours from the Police Stationing San Pedro, Ambergris Caye.

I thoughI had to get one in the states and bring it in to Belize , good thing asked

The Police Report we obtained here has been accepted as a required element of my application for QRP. Would think - but do not know for sure- that the same would apply for Residency.

the police report must be obtained through the Belize Police Dept

FYI....The QRP regulations clearly says "an authentic/police criminal record from the applicants last place of residency". 
You live in Key West the others who answered live in Belize.  Google Belize Tourism Board for the application and list of requirements.