Sorry, I know it's a cardinal mumsnet sin but haven't rtft. I went through a phase of this and have always felt deeply ashamed. When I've thought back over it, there are two important elements that seem to chime with what you've said about your son.
Firstly, as other posters have observed, I felt deprived in comparison to peers at school - 3 sibs and parents were pretty tight on money. No pocket money, had to ask and I hated asking for anything that I actually wanted or needed. Of course school was in a posh area with lots of rich school kids, so not having money for the coke machine or the sweet shop felt very hard. I felt envious of the other kids (who didn't notice or give a shit) and deprived me angry that my mum didn't notice or think about me in this situation - or belittled my feelings if I tried to verbalise them.
Secondly, looking back, what I wanted and needed was more time and attention from my mum and I felt my mum was too over stretched and over stressed looking after all of us and working. I was really ne'er a lot of pressure (from myself but from my mum) to be good and to achieve academically. Which I did in spades but never was praised for - just criticised if my marks weren't up to scratch. I really could have done with a mum that sat down, saw me, noticed me hover on the edge of an eating disorder, noticed I was depressed, just spent time with me and gave me a sense of valuing me for me, rather than for an older child that could look after younger kids, or get good grades, or cause her less stress than my older sib etc.
This second bit of my experience is chiming with what you've said about your family and your son - you have a child with SN in a wheelchair, on oxygen, which must be a child that takes priority in terms of their needs and your time and attention. If your DS has been bullied and still is struggling emotionally, my guess would be his anger and the stealing are to get at your attention and really at positive attention and love. Notice me.
Actions have consequences and punishments are a natural part of this - if he steals in later life, he will face criminal consequences etc. However, from my experience (and I was never caught or punished, just struggled through), I've never stolen aside from this and I hold down a very respectable, professional job.
But alongside holding the boundary of punishing stealing, he needs to get a sense of getting the 'good stuff' directly from you instead of resorting to stealing it. So 1:1 time wi you and his dad where he is able to be listened to and have some of the parenting he sees his sib getting.
He will love that at his age! Grunt swear go away mum. If my mum had offered me this, I would have had no idea how to handle it at all, but there will always be a bit of me that is 12 and desperate for my mum to notice how bad I was feeling and to magically make me feel better. Plus pocket money so I didn't feel such a loser at school.
Sorry, that was long but I hope it helps.