I'd prefer "that".
This is the question of restrictive/non-restrictive relative clause. Restrictive relative clauses don't have commas before the clause, and tend to use "that", though this 'rule' is not as strict as some grammar books suggest.
In this case you do have a restrictive clause, you identify and determine the document you want to be produced as one that has a signature (and not one that isn't signed).
If you removed the clause you would change the intended meaning
... Please produce a document.
The intended limits on the noun "document" are no more.
So I'd prefer no comma and "that" for this restrictive relative clause.