Do they need to be read in order? My library has everything but the first book, and I wasn't sure whether I should put in an ILL request for that before attempting any of the others.

Yes, I'd recommend that. There is a clear dynamic to the relationship between Cormoran Strike and his assistant (Robin Ellacott) -- in addition to which, both of their back stories are gradually revealed over the course of the three books, so by reading books 2 or 3 first you might learn something that Rowling had actually been purposely holding back initially.

I love Robin. I'm glad she continues to appear in the series!