No, the Contract ID is for your org's corporate account. Please note that it's possible that there may be more than one Contract ID per corporate account. Your LinkedIn Recruiter rep should be able to help you locate your Contract ID(s) if you're having trouble.
As part of enabling the integration, Greenhouse will connect each existing Recruiter seat with individual Greenhouse users. Should you add Recruiter seats at any time after the initial configuration of your integration, just send a note to email@example.com, and we'll make sure to connect the new seat to the appropriate Greenhouse user.