Microsoft recently announced the GA for "Bridge to Kubernetes" which enables hybrid development on a kubernetes cluster. Does it also support on-premise kubernetes cluster (without an Internet connection)?
As you can read here:
Bridge to Kubernetes is expanding support to any Kubernetes. Whether you’re connecting to your development cluster running in the cloud, or to your local Kubernetes cluster, Bridge to Kubernetes is available for your end-to-end debugging scenarios.
Support for Bridge to Kubernetes on any Kubernetes cluster is initially available in the VS Code experience and soon after in Visual Studio.
So yes, you can use it also with your local kubernetes environment.
More on its usage and configuration you can read in Microsoft's official docs.
You may also take a look at Use Bridge to Kubernetes with Visual Studio Code.