Use a (mechanical – not electronic) Shabbos clock to set it to go on at specific times, and fill the cup before the machine goes on. If absolutely necessary, you can also change the times it will go on and off on Shabbos (so long as you don’t make it go on or off right then), as that is Grama, which is a Derabanan and is allowed in the case of a Choleh.
This is what we were told when we had to use one for one of our children – but you should confirm this with your LOR yourself.
Good luck, and a Refu’a Sheleima to whomever needs the nebulizer (my kids outgrew the need as they got older, B”H).
an Israeli Yid