[icedchai]

Do you happen to know all of the hostnames / subdomains? Or can you scrape that from their web UI? Without the ability to transfer the zone or get the actual zone file, perhaps you can "export" each record with something like a "dig -t ANY host.example.com" for each record. Not fool proof, but better than nothing...

[domain-tempacct]

hostnames: yes. subdomains and similar things like txt records: no.

[icedchai]

How many records are we talking about? Can you view them as a list in a web interface? If so and given no other alternatives, I’d try to scrape it out of their web UI.

[domain-tempacct]

about 50. no.

[icedchai]

sounds like a pain! sorry your registrar isn't cooperating.