You're welcome.

The CNA listings are on the MITRE website as you noticed. Need to look at the Participating CNAs link there to see which organizations are included. If the application vendor is a CNA, you MUST apply through them directly. https://cve.mitre.org/cve/request_id.html#cna_participants

As for determining if a CVE ID is issued already or not, that requires research. Google searching the application, checking out MITRE's listings, etc.