Is the DayPilot javascript library compatible with Salesforce Lightning Web Components? Specifically looking at the scheduler functionality.
It should be no problem.
You'll need to follow this Salesforce guide for using third-party libraries: https://developer.salesforce.com/docs/component-library/documentation/en/lwc/lwc.js_third_party_library
There is also a tutorial available that goes through the configuration step by step and includes a sample Salesforce project with the Scheduler component for download:
Using JavaScript Scheduler in Salesforce (LWC)