Begin at the top
A missing cap leaves the flue open to rain. A damaged crown lets water enter the masonry. Photographs from safe roof access help separate those two problems.
Look at joints and flashing
Open mortar and gaps where the chimney meets the roof are other possible entry points. A surface treatment is not a substitute for fixing those openings.
Ask what the repair addresses
The quote should identify the failed part and what will be repaired. Request after-photos of any area you cannot see from the ground.
This guide explains the service. The appliance instructions, site conditions and applicable local requirements determine the work.