Shockingly, no. A few months ago I started registering new domains with them, and was very surprised that you had to buy credits for your account instead of having an auto-renew.

I went with name.com instead when moving from Godaddy today. Name.com is about the same price as Namecheap, and they do appear to have automatic billing. The only nicer thing about Namecheap was the cheap SSL certs you could get with a domain purchase.