on-premise bridge to kubernetes
As you can read here:
Bridge to Kubernetes is expanding support to any Kubernetes. Whetheryou’re connecting to your development cluster running in the cloud, orto your local Kubernetes cluster, Bridge to Kubernetes is availablefor your end-to-end debugging scenarios.
Support for Bridge to Kubernetes on any Kubernetes cluster isinitially available in the VS Code experience and soon after in VisualStudio.
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.