My 5 year old has allergies, is it normal for her to cough a lot at bedtime every day?

Yes: Night cough also could be sign of asthma. See a doctor, do allergy test and spirometry.

Don't let her drink: She may be having some reflux. Often parents let their children eat or drink right before going to bed which can be the cause this cough. Don't let her eat or drink anything for at least 90 minutes before laying down. That should help.

Allergens in bedroom: There may be something in the bedroom that she is allergic to, such as stuffed animals or bedding.