Can a Chairman of a Committee vote in a 'Vote of No Confidence' against him in a Executive Meeting of an non profit making organisation?

Assuming nothing in your organization's by-laws preclude it, Robert's Rules of Order recognize that a chair is technically a member of the committee and, as such, has the same voting rights as other members. Traditionally, in larger organizations, the chair normally only votes to break a tie, but that is a matter of courtesy rather than the rules.

Normally, on a confidence motion, the chair should allow that portion of the meeting to be run by the vice-chair.

There is no such thing as a vote of No Confidence. The vote is a Vote of Confidence. Should it be lost, the members have shown no confidence in the person.

Your case- look to the rule book.