Moving with young children.... any advice is welcomed

This may not be of true help But I can say Although there are not any  specific care or educational  plans devoted to Autism in Belize, my Autistic grandson absolutely thrives when staying in Belize. I think it is the lack of that over stimulation  just about everywhere here in the States, He quickly relaxes and we have very few 'Behavours' and he eats and sleeps so much better.  His Asthma problems are almost zero, whereas in California he has to take lots of  daily medcines. I think the lack of Processed food and traffic exhaust pollution helps in this. His non autistic Sister loves Belize also. These kids are older than yours, but their parents agree they wish the children had started coming to Belize years ago as they get so much out of it.  They were down for spring break and are already asking when they can come  during the summer.

Thank you for your comment.  My son is intelligent and very attached to me.  He has not progressed well in all the schooling and therapies I have him in over the past 2 years (includes Applied Behavior Analysis ABA therapies, Occupation Therapies, Speech/language therapies).  The only time I feel I can get him more focused is when we are visiting my parents who live out in the country.  He is a different child when taken away from lots of crowds, noises and distractions.  He is intelligent though communication is a challenge.  He just loves to be outdoors and the winter months here are awful on him and me since he can't get outdoors much.  At this point (I have multiple degrees and consider myself fairly intelligent, in addition have taken multiple training courses in ABA therapy) I think I can homeschool him better.  Only progress he made was when I put my career on hold and worked with him full time.  When he went back to full time therapy and schooling (not with me at a great facility) he only regressed.  I just feel I am in a place in my life that a drastic change maybe the best for my children, specifically my son.  My daughter is extremely advanced for her age and I think she will excel no matter where she is so I don't have too much concern for her.  I do have a concern on making friends and having a social circle for my children to benefit from play dates while down there (socialization is important for them both, especially my daughter). 
I have a long list of things I need to check into still before I make the plunge. 

Such as :
Pediatric care - where to take them if they do get ear infection
Pharmacy - my son does take 2 medications (a sleep aid is one of them) and my daughter does have mild asthma issues here
Drive/fly/ship items there.
I'd like a car though from my experience I don't think my 2013 Honda fit is the best vehicle to have down there if do any road tripping a 4x4 probably be best.  Car insurance?
Schooling for my daughter so can get socialization and I know there really is no schools for children with autism down there so I'd prefer to continue his education through homeschooling.
Tourist visa to residency to duel citizenship process.
Big one is housing... where to move (I always loved the quiet beauty of caye caulker and no need of car is a plus) - love a long term rental until ready to buy. Safe area where there are other families.
How easy to get internet service as I will be putting in 10 hours a week remotely.
Is rental insurance a thing there? Is health insurance needed or is pay for what you need/use?
Good dentist in case my kids need it or will I need to fly them back to the states each time.
So many things to look into. Eventually I will get things figured out and hopefully be moving there.

Thanks for the wonderful reply - give me some encouragement that my son will thrive there.

The problem with cars is import duties are very high and local used cars are usually junk.  Maybe buy new if you can but a car down there may not be able to be brought into the US. Rent for the first while till you learn the lay of the land.  Contact the embassy RE: your sons autism, they have district volunteer wardens who know the lay of the ground.  Katie Valk is one such warden and is a local travel agent   http://www.belize-trips.com/
"Stan Creek district or Caye Caulker. "   neither makes sense.  Being near Belize City or the northern border will provide the best access to health care.  There is a great school near Belmopan.   Southern Belize is much rainier than north belize. I dont know if you equate the autism with vaccs but they are a very serious issue in a tropical climate.

"I do have a concern on making friends and having a social circle for my children to benefit from play dates while down there (socialization is important for them both, especially my daughter).  "  there are children in Belize.

Your over arching issue is can you be patient with a third world country.

oops don't know what happened there sorry.