Is it asthma or allergies when you cough on the soccer field?

Either or both... A cough occurring during exercise could be the result of bronchospasm and this may or may not be related to allergies. It would need to be evaluated by a doctor so diagnosis can be made and treatment given.
Asthma. Cough, particularly with exertion, is a common symptom of asthma, and will respond to your albuterol inhaler. Asthma is often due to, or made worse by, allergic reactions to airborne allergens such as pollen. Allergies can also cause a cough, by leading to significant postnasal drainage that irritates the throat and triggers a cough to get rid of it. Albuterol does not help this type of cough.
Could be both. This could be exercise triggered asthma and / or allergy related asthma. Your allergist can help you figure it out and come up with a good treatment plan.