First start your vpn connection and then restart the emulator, now the emulator should use the host vpn connection.
it’s mostly because of the DNS issue, according to the android doc:
At startup, the emulator reads the list of DNS servers that your system is currently using. It then stores the IP addresses of up to four servers on this list and sets up aliases to them on the emulated addresses 10.0.2.3, 10.0.2.4, 10.0.2.5 and 10.0.2.6 as needed.