I'm sure 椅子 is more common but I can't find any information on the nuance of 席
The difference here is very much like the difference between
seating in English.
席 is a slightly more abstract term primarily used to mean "somewhere to sit", in the same way
seating is used in English.
席 could be a chair, or a bench, or a cushion on the ground. Phrases such as:
この席、空いていますか？ (Is this seat open?)
席が足りません。 (There isn't enough seating.)
Are fairly common, and while you might be understood even if you replaced
椅子, it would sound very literal - you'll just be talking about chair(s).
Note that there is also a word
座席, which is quite similar to
席 with minor differences as described here.