I'm an adult with special needs and I want to move in with a family who I'm really close to. How do I go about doing this?

Have an open conversation with your friend about how they would feel about that.

You say you want to move in with them, but do THEY want you to move in with them?

Before you can go any further, you have to make sure that this family WANTS you to move in with them, and is WILLING for you to move in with them. 

Many families don't want to take on the responsibility of having someone with "special needs" move in with them. 

They might be really close to you and you might be really close to them, but that doesn't mean they want you to move in with them.

You cannot move anywhere now with the virus in the world, you have to stay where you are.

just ask them about it

Ask the head of that household if they can help you out and if you can join their family. That's the only way to do it.

Did they ASK YOU to? I'm guessing they did not, otherwise you wouldn't be asking this. You have to build on your own relationships & family life, not scab off others you envy.

You hadn’t mentioned if it was proposed to you, and that is the main factor.

If its what you want, and its what they want, you dont need a piece of paper to give you permission, you can just do it.

Have you been invited?  If so, call a moving van, pack up and move.

In The USA, even though you are an adult and not related you can be claimed as a defendant on your host’s taxes.  I don’t know if GB has such a thing.  If you’re receiving government benefits you could pay some form of rent to your benefactors to help with expenses.  

You need to know if they want it with it, first. Your status would be as a lodger, but there are things that need to be discussed beforehand.