At aged 5 and a half it is unreasonable to expect the child to administer their own inhaler. It is the teacher's responsibility to help the child to use the inhaler as asked in the forms/ asthma card filled in and signed by the parent. If the teacher is unfamiliar with using an inhaler, the parent must demonstrate. If the school doesn't meet the child's medical needs, then they are failing in their duty of care.
Our school has bought a spare inhaler for use when a child needs their inhaler and for whatever reason it is not available. All our class teachers are responsible for any inhalers for the children in their class and they have to be packed and brought on any visits etc.
I hope the situation has now been rectified!