My 6 year old daughter has mild asthma and is a competitive swimmer. would an inhaler before swimming help her not get short of breath in the water?

Most likely yes: 90% of patients with asthma will have symptoms that are exacerbated by exercise. In those patients I have them take 2 puffs of their albuterol inhaler 15-20 minutes prior to exercise in an attempt to prevent the asthma symptoms.

Yes: Sure thing. Also recommended, short warm up followed by a 10 minutes rest period prior to exercise.