Each Local Authority should have a list of libraries within their Council area. These may be static i.e. located in one building all of the time, or mobile i.e. they are in vehicles that attend a set location on a specific day at a certain time. This data may also be collected as part of other datasets (e.g. Council Asset Register) though Local Authorities do appear to hold it as a distinct layer.
Further information on Libraries in Scotland (inc. non-LA libraries) is available from The Scottish Library and Information Council (https://scottishlibraries.org/)