My autistic son will not leave the house/go outside - can anyone help?

Hi @Eloner sounds like you have support online at least which is terrific.  Such a relief to know other parents understand situation of parents in similar position. I am one of those.  I have 2 teenage sons on autism spectrum, high functioning but with loads of anxiety.  When I read your news and that of others I cried to think other people are experiencing the same difficulties and there are other children like mine experiencing difficulties.  
My husband and I have struggled through many years, with our children only diagnosed at 14 and 12.  It took 3 years to obtain diagnosis as my eldest son would not attend the medical sessions.   It has not been any easier since, as anxiety has persisted.  Very little help from anyone since. That said, our life is not all bad, even though I can identify what my children do as being the same as your son.  There is NDIS in Australia now which is helpful.
I have tried homeschooling in various forms along with trying to have my children return to school, with a lot of failures in my husband and my eyes but now achieving some breakthroughs.  I have taken all stress away and said they can just learn to cook starting with toaster, microwave, heating in oven, and soon the slow cooker and then cooking a roast in oven as well hopefully. Next will be some gardening when I make the garden neat again as I discovered my eldest son does not like a messy backyard and wants green grass.  I said fine as long as he mows it!  They both play never ending computers with online friends which I allow during night during school holidays only now.  I encourage a relaxed homeschooling program other times.  They do have some friends visit occasionally who were neighbours.  My children are starting to take care of their dress codes and appearance now.  Slow progress but such a delight!  So I am hopeful what the future will bring but nervous at the same time.  They stay indoors without contact in person of friends.  They have us though and I love them dearly with and without their challenges.
Re work my husband and I have struggled through various moments of my husband's employment and unemployment in parts all the while I struggled running a business while homeschooling.  Best thing we did was for my husband to work full time and I struggled through with my business which is a profession, as I can now offer my children a job in administration and who knows what the world will bring.  It is a starting point and I can see happiness and achievement now and around the corner. 
Family support has been listening to us but on the whole they have proven to be very ignorant in views and understanding of autism.  So the struggle to be heard and understood for my children and us as a family is stressful.  Nevertheless I have hope and I have realised the happier I am as a parent the happier my children are, as is any child.  So trying my best to reach out (not my usual way) to others in same position for understanding and support.  Hopefully I have given you some hope as you have made me realise there is support out there and lovely people to connect with.  Enjoy your life with your child and friends and family.