I also have Hashimoto's. Like Tigger714's husband my numbers were so high they couldn't believe I was still upright--single working mom--of course I can still function with a disease ravaging my body--I am WOMAN
Anyway---I agree with everyone else--get her on meds ASAP. Also--get a copy of her test results and fax them to the endo's office and explain the URGENT need. Quite frankly--if they can/t/won't accomodate you with her numbers THAT hight I would shop for another endo. You could also try calling every day once in the morning and once early afternoon to see if they have any cancellations. Another tactic might be to send her to your GP (not her ped). He/She may be more willing to assist you.
As far as the trip:
keep hydrated (water water water)
Don't plan on commando style touring--you can plan a mix of rides/attractions with the more "restful" ones in between (Hall of Presidents, Railroad, Philharmagic, Universe Of Energy, American Adventure, etc. do a search on the boards for good 'resting" attractions.) You can do this without her actually being aware of it if that would be an issue.
I would plan your day in "blocks" (which we often do). An example would be Morning block: 9:00 - 12:00 Park; afternoon block 1:00ish - 4:00ish--pool, rest, etc 5:00 ish-night park
ICE PACKS--I seem to be able to tolerate cold better than heat. I buy the disposable packs in the first aid section (about $1.25) and always keep one on me.
Light colored umbrella for waiting for outside shows and parades
keep on a meal/snack schedule. Try not to let any dips/surges take place.
Bring some stool softener/Miralax along. Constipation can be compounded by all of the "park food"/not keeping to regular eating schedule. Not a laxative--you don't want her in the run to the restroom mode--but just something to help.
Just watch her and when she seems to be flagging (sometimes you will be able to tell by attitude, speech, etc not just physical symptoms) take a break.
If all else fails and she is overly tired--stay at the hotel and lounge by the pool. Plan activities that you know will be in short duration but she would still enjoy (a few hours at DisneyQuest, mini golf, water mice, afternoon tea at the Grand Floridian, a movie at Downtown Disney, etc.).
You didn't mention your party size but if she is passing out and having seizures I would not leave her alone. You may need to split up from time to time so someone can 'rest" with her.
Along the same lines--be careful of rides that may make her dizzy.
HTH
As a side note and unrelated to Disney---I notice she is 16 which is getting ready to drive time--if she has had a seizure then in SOME states she may be unable to get a driver license. I know many many many years ago in MD--you had to be seizure free for THREE years to renew/get back/obtain a license. I was lucky that I lived close to so many family members and my job!!!!!!! I also had to resort to using public transit--UGGG!!!!! I'm just not a public transit type of girl with all those other people and germs!!!