If I hired a web designer to do my website, do I have any recourse to receive a refund if they did not live up to their contract?

If the web designer was contracted to do something and didn’t do it, then you may be due something as a breach of contract.  However, that doesn’t mean that each and every little detail entitles you, for example, to a full refund.  The legality of it has to do with whether the breach was material or not.  Of course, really it all comes down to the contract.  Web design is a combination of creative art and technological skill; therefore, it has started to become a hot topic for breach of contract.  Have your contract reviewed by an attorney and discuss the issues you have with the site.  That will be the way to determine your legal standing, figure out your options, and decide how to proceed.