Do not wait until tomorrow. Take him to the doctor today and if his pediatrician says he can't be seen today, I'd go to the ER now. The fact that the other arm is now swelling is not good and is a strong sign he needs to be seen now, without waiting; something is spreading. It could be an allergic reaction but it also could be an infection -- there are lymph nodes in the arms (both under the armpits and near the elbows) and they can swell if you have infection not just in the arms but elsewhere in the body; you cannot know what is going on yourself and really need him to be seen now, not tomorrow!
I had a swollen arm recently and it turned out I had an infection in my finger (in a very, very tiny cut--I had barely noticed it) and it still was enough to swell that arm painfully. An untreated infection will spread through the body.
This is nothing to do with your housekeeping; that doesn't matter. I sounds like you're assuming this is a bug bite and trying to figure out how he got bitten but don't worry about the "how" right now. Please get him seen today and update us here. It likely is something minor but it's time for a doctor.