Thank you for the push in the right direction! I was able to solve the issue with this code

os.environ["DATABRICKS_CLIENT_ID"] = ""
os.environ["DATABRICKS_CLIENT_SECRET"] = ""
os.environ["DATABRICKS_TOKEN"] = os.environ.get("VAR_CONFIGURED_WITH_DATABRICKS_SECRETS")

# Enables trace logging by default
mlflow.set_tracking_uri("databricks")
mlflow.set_experiment("/Users/my-name@mycompany.com/my-experiment")
mlflow.openai.autolog()

Note that I did not need to set DATABRICKS_HOST, as that's already set in the App's default environment.

Thanks @samshifflett46 !