I've been trying to get cloudflare dynamic DNS to work on a new VyOS installation. To troubleshoot I ran ddclient direct with debug messaging and got the following result.
RECEIVE: HTTP/1.1 410 Gone RECEIVE: Date: Tue, 20 Nov 2018 21:40:57 GMT RECEIVE: Content-Type: text/html RECEIVE: Connection: close RECEIVE: Set-Cookie: __cfduid=d53c907651da69ff19bfc787be4f74ac11542750057; expires=Wed, 20-Nov-19 21:40:57 GMT; path=/; domain=.cloudflare.com; HttpOnly RECEIVE: Cache-Control: public, max-age=31536000 RECEIVE: Set-Cookie: __cflb=1594113181; path=/; expires=Wed, 21-Nov-18 20:40:57 GMT RECEIVE: Strict-Transport-Security: max-age=15780000; includeSubDomains RECEIVE: Expect-CT: max-age=604800, report-uri="https://report-uri.cloudflare.com/cdn-cgi/beacon/expect-ct" RECEIVE: Server: cloudflare RECEIVE: CF-RAY: 47ce13f459c8ab90-YYZ RECEIVE: RECEIVE: This API has been deprecated in favor of API v4, available at https://api.cloudflare.com
It appears the ddclient version used in VyOS (latest rolling version) does not support the current CloudFlare API.