I agree with Tom. My husband runs a business very reliant on subcontractor performance and so there are strict policies and procedures in place that essentially outline how paperwork must be handled before a sub gets paid. Simple as that. Sounds like it may be time to create a Standard Operating Procedures document (SOP). It's tedious at first but eventually it becomes second nature to everyone.
Also, be prepared for a lot of resistance in the beginning. Some may even leave your company on account of your new SOPs. That's fine. You have a business to run, not a daycare

And any new talent you take on will come to know your SOPs as a matter of course in doing business with you. They will respect your high standards and know that such processes are how you keep your business sustainable and in turn protect the best interests of your clients AND subcontractors (i.e. no business means no work for them either).
HTH,
Karri