How can I get a access to the nuts and bolts of a contract to review performance of the contractor/vendor? Between a 501c3 organizations an LLC

Why don't you have a copy of the contract? What nuts and bolts? Everything is in the contract or it isn't. What is your position in this situation?

If the not for profit hired a company to do a job and didn't specify all that was required, then they made a mistake in not doing so, and their next step is consulting a lawyer. But your question is too vague.