I prefer party to a contract or agreement than contractual party (which sounds too literal). A person or organization is a party to a contract or to an agreement. This is standard legal phrasing. I hope this helps.

That is, the parties that signed the original order agreement.

Since this refers to an order, I think it would be an "agreement" instead of a "contract" since "order agreement" is an often-used term. (I like this better than the last idea I sent.)

. . . strikes me as most appropriate, unoriginal though it may be. As I read the excerpt, there seem to be three entities involved: the party placing the order, the part receiving the order (the Societe XXXX), and a third group consisting of the other companies or individuals with whom the Societe XXXX has contractual relationships. In other words, "The order cannot be processed without the requested information, which may be disclosed to XXXX's contractual partners." You might say "affiliates" or "associates" instead of "partners," but "partenaire" is perfectly acceptable.