Can babies as young as 1 get allergies?

Certainly. Foods are more common in this age group, particularly dairy and eggs.
Certainly. Food allergies may play an important role in aggravating atopic dermatitis in addition to inducing classic gastrointestinal problem such as diarrhea. Indoor allergens, e.g., mold and dust mite also trigger allergic rhinitis.
Yes, but. Most of the congestion in babies is due to airborne irritants.Dust,cleaning fumes,fragrances,hair spray,air fresheners, etc. are more problems for small babies with relatively small noses. Many insurers won't cover testing for airborne allergies under the age of 4. On the other hand, selective food allergies are common, especially in kids with a skin condition called eczema.