The website for a domain that my organization previously owned is apparently cached on Cloudflare. We would like to contact the site owner to discuss purchasing the domain and/or decomissioning the website since it contains outdated information.
Welcome to the Cloudflare Community. 
Cloudflare does not have the ability to contact the new owners of a domain on your behalf. However, you can use WHOIS lookup tools available online to find the contact information of the domain owner. Alternatively, you can reach out to the domain registrar for assistance since they might have the contact information for the current owner. Any site content, including outdated information, is under the control of the site owner. You can discuss decommissioning the site or purchasing the domain directly with them.