Hair clippers, like hair dryers and showers, are apparently the work of the devil here and must be avoided at all costs. I know a DC who can only have their hair trimmed by a close family member, another uses a home visiting hairdresser, another only in their sleep, and another not even then - after trying everything, the parents have had to give up, at least for now, and embrace hats, hoods and the permanent bed hair look.
For us, it's taken years. The first person we tried a few times was a disaster when WB was very young - she was too insensitive and excitable, so Lord knows how she'd cope with him now and we her. Our second person turned out to be a star - very unexpectedly - as she is very trendy and not even a parent herself, let alone of a DC like WB. We've stuck with her ever since, just persevered though it hasn't been easy, gradually developed a routine, and it (more or less!) works.
Tips are:
Find a calm, sympathetic and understanding person who will be willing to be flexible and take it as slowly (or as quickly!) as necessary.
Explain fully what you need and why, and what the issues and problems are likely to be.
Book well in advance to secure the quietest times - for us, it's when no one else is in, first or last thing.
Avoid at all costs being in a busy, noisy, salon.
Take your DC's booster chair, houdini harness, wheelchair, or SN buggy if needed - they can even sit on your knee, it doesn't have to be in the hairdresser's chair.
Take sensory distractions - drinks, snacks, fiddle toys, portable DVD player, iPad, whatever.
Don't make your DC wear a hairdresser's cape - it might come in time, but it took years for WB to wear one, so we just accepted hair on clothes.
Don't make your DC have their hair washed first - WB can't, even now.
Ask in advance for no hairdryers or loud music on while you're there, remind them if necessary, or ask for your DC's to be played, (if not using earphones or ear defenders).
Use pictures like PECS beforehand and take photos while there for future reference and as part of your DC's social story.
Stand alongside your DC and the hairdresser to provide reassurance (to both!), keep the distractions mentioned above coming, preempt or deal with any meltdowns and - very important, this - keep hands away from scissors.
Try to remain calm and keep your sense of humour - not always easy, I know.
If you find 'a star', as we've done, show your undying appreciation regularly - these people are rare in life and worth their weight in gold - you might even gain a friend.
And remember, if you have to leave after half a haircut - as we have done - you can go back another time for the rest. In the meantime, enjoy your DC being mistaken for a young member of Human League.
HTH.