How to inject Vault secrets in a common Helm chart in Kubernetes?

Consider this helm chart for mlflow

How can I install it with injecting secrets for a PostgreSQL cluster and AWS S3 bucket credentials?

The Vault secrets could look something like this

kubectl exec -it vault-0 -n vault -- sh -c "
  vault kv put internal/mlflow \
    DATABASE_URL=\"$PRIMARY_DATABASE_URL\" \
    PGUSER=\"$PSQL_USER_NAME\" \\
    PGPASSWORD=\"$PSQL_USER_PASSWORD\" \\
    AWS_ACCESS_KEY_ID=\"$AWS_ACCESS_KEY_ID\" \
    AWS_SECRET_ACCESS_KEY=\"$AWS_SECRET_ACCESS_KEY\" \
  
  vault kv metadata put -max-versions=$MAX_VERSIONS internal/mlflow

  vault policy write mlflow - <<-EOF
path \"internal/data/mlflow\" {
  capabilities = [\"read\"]
}
EOF
  
  vault write auth/k8s/role/mlflow \
    bound_service_account_names=mlflow-server \
    bound_service_account_namespaces=mlflow \
    policies=mlflow
"