Is it normal for toddler to cough after running if they have allergies or cold?

No: Cough with exercise, especially during an allergy attack or a URI suggests asthma. This can be treated, see your dictor.

Yes: But if the coughing hinders ability to play or cough does not resolve shortly after resting then need to consider possibility of reactive airways disease.

Common; not normal: It is quite common to see children experiencing coughing with activities/allergies etc...But this is not normal. If allergies causing the cough, then treating the allergies; same thing here, allergy is common but not normal. Coughing with activities suggests possibility of exercise-induced asthma. If this is persisting, consult doc. Good luck.