I am interested in finding the ISI-indexed papers for a given institition X. This would be any ISI-indexed paper where one or more authors are affiliated with X. Is this doable and are there any guidelines for how to do this?
The easiest way would be to do a Web of Science search using the 'Address' field, with the university name or institution name as query. This will most likely return lots of results, and you will maybe need to go through them manually to do some quality checking. You can also refine search results by 'Countries/Territories' to filter irrelevant results (e.g. similarly named institutions in different countries). You can also use the 'Address' field to search for individual department names while also including city and country for some good results.
You can ofcourse do similar searches in Scopus etc. as well, but if you are specifically after ISI-indexed papers you should use Web of Science (maybe even restricting yourself to only the Web of Science Core Collection database).
Ask a librarian at Institution X. You would probably get your answer in a few minutes or maybe in at most 24 hours.