My son too is asthmatic but rarely needs his nebulizer anymore (praying that he is 'growing out of it!!') I usually take his nebulizer on trips, but the last time I flew, the baggage people almost broke it --trying to inspect it, I guess. I talked to ds's pediatrician and he said since he rarely uses it, to take his inhaler instead. The inhaler would be good enough for a couple days until we got home, he said. If the child got really sick, I'm sure the dr. onboard could help.
So, maybe the mom should ask the child's doctor.
Also for the mother to consider is what causes his asthma. For my son it is pollen, seasonal allergies, hay and mold. Out in the middle of the ocean is a VERY healthy place for him. If the child has exercise induced asthma, he may well need his nebulizer with all the running, swimming, snorkeling.