grammar - do anything vs do something

They are both correct. They only differ in meaning.

The first one, "Please let us know if we need to do anything about it," means "You paid for the chair; we delivered it. Hopefully there's nothing at all wrong with it, so please don't call us unless you want to buy something else. Even if you really, really want to call us, please, just bite the bullet and don't call us. Desist. Control yourselves."

The second one, "Please let us know if we need to do something about it," means "That chair we delivered to you the other day is missing a leg. We know because you've already called us about it three times. We hope you don't mind. You don't mind, do you? It's just a chair after all. It's not that big a deal. However, if you really insist and keep annoying us with your calls and complaining, and threaten to sue and all, we might actually consider fixing it, or giving you some of your money back. We would still have to charge you for the original delivery, as well as for taking the chair back, so it's probably not even worth it, but, well, if you insist on being a pain in the ass, yeah, we'll swing by your house at 4 p.m. tomorrow or next Tuesday and give you an assessment and an estimate."

So, yeah, the first version is better: it's a lot cleaner, and might save you a whole lot of trouble.