Do I have to agree to allow a tenant who is notifying me of intent to break lease agreement find a replacement tenant? The lease does not have a clause allowing tenant replacement.
You do not have to accept the tenant they bring you, but if they move out you have a legal obligation to mitigate (minimize) your losses. If tenant brings you a qualified applicant, and you turn him down, they can claim you are not working to mitigate your loss. That's why written screening criteria is recommended. When a tenant states they want to break the lease, your first obligation is to get it rerented ASAP.
I agree with @Andrew B. , you need to make a good faith effort, but not to take whoever they get. Treat their "find" as any other applicant that needs to meet your criteria.