Is it okay to give albuterol thru nebulizer to my youngest child (5yrs) who has been coughing daily?

Need maintenance Rx. Nebulizers are good, but with a little practice, 5 year olds should do well with a metered dose inhaler (mdi, often called a puffer) & spacer. (spacers = large "tubes" to use between the child & the mdi; they help make sure the medicine gets into the lungs where it helps.) mdis are much easier + kids can gain some independence. But, daily albuterol need means you need maintenance rx. See your doc.
As prescribed by Dr. A cardinal symptom of asthma in kids is coughing. Some children with asthma do not "wheeze" and they mainly cough with activities, or if they laugh to hard, or they run up stairs, sometimes your symptoms are worse at night. Asthma is a combinatiion of chronic inflammation and acute spasm of the airway. The albuterol helps the airway open up. The nebulizer is a great way to deliver the albuterol.
Remember Dr.'s plan. A 5 year-old child who needed albuterol oral inhalers or albuterol nebulizer treatments in the past for coughing or wheezing, may need the same treatments for similar symptoms in the future. The doctor will have given instructions to the parents on "home use" of the inhalers and nebulizer. Parents should call the doctor if unclear on instructions. A child should not use somebody else's medicine.