On no account
"On no account" is most often used either to prohibit something ("You should on no account do X") or to state intention ("On no account will I tolerate X"). It is very rarely used to describe past events: this may be why we would be unlikely to say "On no account did he commit the crime" or "He on no account committed the crime". It would make more sense to say "On no account would he have committed the crime", surmising that there were no circumstances under which he would have done so.
We could say "You should not sign the contract on any account" or (better) "You should on no account sign the contract", "On no account should you sign the contract".
Don't use a double negative with 'on no account'
In standard English, we would never say "You shouldn't on no account sign the contract" with a double negative. Even in non-standard English, it is an unlikely sentence because it combines the relatively formal sounding "on no account" with the informality of double negation.
Double negation in general
You posited that "I don't have no money" is an emphatic way of saying "I don't have money". It isn't. In dialect and very colloquial usage, "I don't have no money" means exactly the same thing to most speakers as "I don't have any money" and is no more emphatic.
(Also, it is important to note that using a double negative like this is considered non-standard and would be avoided by some speakers even in the most informal scenarios. Further, in standard English "I don't have no money" could be used - and occasionally is used - to mean "I do have some money".)
Still, your comparison was between "I don't have no money" and "I don't have money" (without "any") - and "I don't have any money" is probably slightly more emphatic than "I don't have money" (since the latter could be interpreted as meaning "I have a little money but not enough"). The usual wording is "I don't have any money" (or "I have no money") - we rarely miss out the "any", unless perhaps we want to suggest that we have some (but not enough).