Your SQL Server Express must NOT be installed as LocalDB (https://docs.microsoft.com/en-us/sql/database-engine/configure-windows/sql-server-express-localdb?view=sql-server-2017)
It has to be installed as database engine
From inside the VMWare, use SQL Tools to connect to your database such as tools from here: https://docs.microsoft.com/en-us/sql/ssms/tutorials/connect-query-sql-server?view=sql-server-2017 or 3rd party tools like this one: https://www.apexsql.com/Download.aspx
You need to know first that the SQL express instance is working. If you cannot connect from within the VMWare appliance itself, you will not be able to connect from outside.
If above is working locally already, check the VMWare config.
To minimize issues, turn off firewall inside the VMWare appliance.
The network configuration cannot be NAT. If it is under NAT, then the VM Appliance can see your host computer, but your host computer cannot see what’s inside the VM Appliance.
So my suggestion is you do a bridge connection. This is much simpler. Then if your network connection is on DHCP, then make the guest connection also DHCP and just make sure you knos the IP. so you can configure to SC. But this will just make things hard.
So I suggest that your guest OS inside VMWare is on Static IP so it doesn’t change very frequently. Better if your network is also on static to minimize issues such as IP conflict.
Re-Use the tools inside the VMWAre appliance, install on your host computer the tools to test whether you can already connect to the SQL Express inside VM using these tools. If still you cannot, then SC cannot connect also.
Once you are able to connect using the tools, then there is already possibility that SC will connect… but then again you will need the correct jet drivers to get the ODBC thing working.
Good luck, you got a lot of troubles ahead.