After discovering that Gengar is one causing all the trouble at the PCL during Chapter 3 of the game, it will be up to Tim and Pikachu to capture it. Unfortunately, since Gengar is a Ghost Type Pokemon, they simply can't catch it with their bare hands. Instead, they will need some type of device to help them get the job done. In order to discover such a device, you will need to talk with the PCL Director, Dorothy, in her office. By doing so, you will learn of a device that might be able to help you, one that should be found in the Storage Room in Building B. On one of the tables in said room, you will Find the Capture Device.
With the Capture Device, you will now be able to catch Gengar. But first, you must Find it. After first running into Gengar in the Library, it will head immediately to the Meeting Room. Knowing this, you will need to head there after obtaining the Capture Device. As you do, the next Fast Action Event will take place, the same that will see you shooting an electro web from the Capture Device in order to catch Gengar. With Gengar apprehended, you will be able to question it to see who it is working with.