Hi folks, I'm experiencing an identical problem but I do have a Vnet integration with ROUTE ALL enabled. Our Function App has outbound restrictions as part of the security requirements (default no outbound unless we explicitly add rules etc.).
When using ZipDeploy with RUN FROM PACKAGE, I'm seeing the task that is supposed to mount the Zip package to a virtual mount is failing, so the code never updates from the last successful deployment. Turning Vnet integration off allows for a successful deployment.
I cannot keep switching vnet integration off each time we need to perform a deployment.
Alternatively a working method is to NOT use RUN FROM PACKAGE and ZipDeploy mostly works (KuduSync.NET runs and extracts the zip contents into wwwroot/ ), looking at the logs I can see it seems the deploy process is able to restart and therefore re-sync triggers, but I still get a superficial failure - i'm assuming because the deploy process on the Functions host is trying to make an API call to get status and fails because outbound is blocked. (There's no documentation i can find to suggest exactly what the deployment process is doing so its a guessing game).
However I'd like to be able to use RUN FROM PACKAGE feature if I can.