The ITVS-500 and ITVS-750 both have a panel on the back held on by screws. Perhaps yours is similar. If not, can you share a picture of the top (open, without record), back, and bottom? Even if it's a wooden box glued together, there's got to be some screws or some other access.

Apple now sells some replacement parts for phones, computers, and displays directly on its website. But they do not list the AppleTV, unfortunately, so you'd have to look for used parts, or generic / third-party replacements. If you want to find a replacement cover, I'd suggest looking on eBay, or on a local marketplace where you live, like Craigslist or Facebook market. You might not find just the cover plate, so look around for any "dead" or malfunctioning devices that are still in good physical condition. Another option could be to purchase a vinyl skin -- basically a sticker that comes cut-to-size -- to put on it and cover the damage. In the picture it looks like there is just peeling and flaking. As long as the plastic cover is still sturdy a skin would cover the damage, and let you pick a favorite design for a fresh, clean look. https://www.decalgirl.com/skins/apple-tv-4th-gen-skins is one website selling these. Just brush or scrape off any dirt, dust, and loose debris from your AppleTV before...