No. The expression you want is probably "sickening himself" as in "By eating so much candy, the child is sickening himself". If you have another sense you are trying to get at, give some more context.
Is "sicking himself" proper grammar?
If he was a dog, and if he was biting himself on the butt, you could say he was sicking himself (but you'd have to spell it siccing).
You're probably looking for "sickening" but that's really a bit clumsy. Before recommending that word I'd want to know how he was sickening himself. There are much better ways to express it.
No, I have no idea what you're trying to say.