Not only can you use your nebulizer at first aid, unless they have changed things they will hold it there for you so you don't have to carry it all day.
This is what we did in Feb 2016. There was a short form to fill out, it took maybe 5 minutes first thing in the morning. DD (15yo) kept her rescue inhaler on her and knew to head to first aid or get CM help if there was a problem and she wasn't with me. Her asthma is mostly exercise induced and hasn't ever compromised her so quickly that the time/distance to first aid would be a danger.
Try to remember to pick it up before you head out of the Park. We forgot overnight once. It was a bit of a comedic adventure tracking it down.