There are a number of ways to do this.
One easy way is to set up preAuth on your domain, that way you can get an authtoken for any user you need to get the calendar of, and then use the REST API as you already now. If you implement your own pre-auth mechanism be careful not to expose/open it to the public as this may have security implications.
https://wiki.zimbra.com/wiki/Preauth get an auth token, then you can use REST!
Another way is using an admin account to get the calendar using SOAP. One way to find out how to do this is take a look at what the zmmailbox command does when you invoke it via the CLI on the server
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zmmailbox -d -z -m info@example.com -t 0 getRestURL "/calendar"
This will dump all SOAP requests to the terminal, so you can implement whatever you need from that in your application.