Which elements of this recipe can be made ahead of time? I assume the vinaigrette, but what about shaving the celery, etc.?

Depending on how far ahead of time you're preparing the ingredients, it's possible you could shave the fennel, celery, and radishes and keep them in cold water until you're ready to use them. The hazelnuts can be toasted ahead of time, and yes the vinaigrette can definitely be made ahead too.