Simply put and in kube terms, you can not.
Environment for linux process is established on process startup, and there are certainly no kube tools that can achieve such goal.
For example, if you make a change to your Deployment (I assume you use it to create pods) it will roll the underlying pods.
Now, that said, there is a really hacky solution reported under Is there a way to change the environment variables of another process in Unix? that involves using GDB
Also, remember that even if you could do that, there is still application logic that would need to watch for such changes instead of, as it usually is now, just evaluate configuration from envs during startup.