While you might not be able to exactly grab a zone file directly you can list the records you have permission to retrieve using the dig command with the following options.
dig +nocmd yourdomain.com any +multiline +noall +answer
Its also possible to find the subdomains of a domain by using the authority.
First do a simple lookup on the domain and see if they have an “authority” section. At the time of writing this it seems wikipedia isn’t giving me an authority section so you may want to try something else as an example.
;; AUTHORITY SECTION: wikipedia.com. 163475 IN NS ns2.wikimedia.org. wikipedia.com. 163475 IN NS ns1.wikimedia.org. wikipedia.com. 163475 IN NS ns0.wikimedia.org.
Then do the following using the authority results in the @ to get a list of all subdomains
dig @ns1.wikimedia.org wikipedia.com axfr