My house was completely destroyed by fire. I'm trying to collect on my personal property that I had in the house, but the insurance company is telling me I need an inventory. Can they do that?

Yes. Whether your policy pays for the replacement or just the actual cash value, the company is only obligated to pay for personal property that you can show you owned at the time of loss. It is a very good idea to keep an up to date inventory in a secure place. Also, to help you remember what you had, it is helpful to take pictures of each room and keep them with your inventory.