It is possible:
The Requestor is given a Profile that allows them to submit Requests, but that profile does not have edit ability of the requests in the Project Request Tab. They basically push the data but then no longer have access to it. You may also do this by making the Request as a Web Form. The Team Member profile can make a request, and review request but not edit them once submitted.
The Request Manager (or "Single Named User") is given a custom Profile that has access to Requests whereas any other person in the system should not be in this profile. Project Managers through Admins have this ability.
Should you need tighter controls, you might be more specific about the workflow in a follow up post if you find that my suggestion doesn't quite work. Let us know!
Steve Thompson | Solutions Consultant
Daptiv