You are sadly mistaken. A private citizen is no more able to ask for a pesticide license than for another persons driver's license. You can ask, but Farmer John doesn't have to even give you an answer. A law enforcement officer can ask to see the license (or anything else, for that matter), but they are not qualified nor empowered to enforce any pesticide regulations. While a farmer is responsible for the outcome of his actions, he damn sure can do what he wants until a governing authority tells him to stop.
Now if a citizen called the cops, went out into the field with his shotgun, intent on not taking "No!" for an answer, the cops would probably get involved real quick, place a call to the dept of Ag, and might stop the farmer if they observed compelling reasons for concern. Almost certainly, Mr. Citizen would get arrested for something, just because it isn't legal to interfere with a businessman doing his business, and waving a shotgun around is generally frowned upon, also. Unless... Mr. Dept of Agriculture came along and told the cops that the citizen was correct to interfere. Then things might take a different direction.