You absolutely can get your son tested and should find a pediatric allergist in your area to do so. My son began showing signs of allergies when I stopped nursing him at 9 months of age. He was congested, had a runny nose and seemed to have a cold that went on forever. For him, it ended up being food, seasonal, pet and chemical allergies. Unfortunately, we only found out after he had a severe asthma attack at 10 1/2 months old and was in ICU and intubated for several days. For your child, it could be a food allergy, seasonal allergies (different ones are active different times of the year), chemical allergies (i.e. pesticides, rug cleaners, etc.), pet allergies, or environmental allergies (dust, mold, etc.).
I don't know that they would do shots for a child that young, but I would definantly take him to an pediatric allergist to have him tested, gather information, and discuss treatment options.
Best of luck to you as you seek your answers.