What is this furniture part, and where can I find one?
That is a barrel nut and bolt. You can find them at most hardware stores, but they are a specialty item, so they may not have the exact ones you need.

You need to take those with you to a hardware store or get the model information of the furniture to identify the size, or calipers to get accurate measurements. I recommend starting by trying to find the make and model of the furniture to contact their customer service.