Is the storing of hay allowed at self storage units?

You should check with the company that operates the storage facility. If you don’t want to “make a big fuss” ask your buddy to call the same place that is being rented & ask like he’s actually wondering if he could do it.
The person will tell them the truth. Find out the answer & then go & complain or not—depending on the issue.
If yes—I’d just ask to move your unit due to the “bad neighbor”. Frankly it shouldn’t smell horrible unless it was wet when they put the hay in there. You are right though—hay is combustible making it a horrible fire hazard. You will get more in here about self storage oakville.