Once you integrate chemicals into the sprinkler lines water, you've just created a high hazard cross connection. Any high hazard cross connection requires a reduced pressure principle device (RPZ) for the backflow connection to the supply. All backflow installation requirements are based on a potential backflow situation whether it ever happens or not.
If you are connected to a public water supply, and even if you're not, ask them their position on adding chemicals into your sprinkler system.
I wouldn't do it for that reason alone. But what about another scenario: Your kiddos, neighbors, or animals happen to be in that mosquito chemicals area and the zone came on by accident and sprayed them? Or the zone stuck open and now your chemical run off is flooding your neighbors lawn, going down the street into the gutter that goes to a local creek?
Just throwing stuff out to think about. I hate mosquitos too!
Have a great day!