You're right, we're not lawyers. But we do have one. And he's kindly written up a recruitment agency agreement template for you to use.
Working with multiple agencies and their agreements most likely means you and your managers are agreeing to many different terms and conditions with many different agencies. It makes things complicated for your team, and let's be honest, agency terms are primarily written to protect the agency.
And we strongly recommend you get your own legal advice on the agreement before you start using it. But it does give you a pretty good head start.