The Constitution does not preclude police forces as it does not prohibit sheriffs or constables which were commonly derived from English traditions. If the Consitution does not specifically address the subject then it is left to the states and/or localities. Without sanctioned law enforcement the citizenry could arrest, detain and/or kill criminals on their own or as groups; would that be better for the criminals or would it mean that they could do what they wanted? It might be more efficient in controlling crime but it is not likely.
That method has been used in the past in many nations in many different ages.