You can use Prometheus to monitor Vault. You can configure Vault to expose metrics through statsd. Both the Helm chart and the Vault Operator installs the Prometheus StatsD exporter and annotates the pods correctly with Prometheus annotations so Prometheus can discover and scrape them. All you have to do is to put the telemetry stanza into your Vault configuration:
telemetry: statsd_address: localhost:9125
You may find the generic Prometheus kubernetes client Go Process runtime monitoring dashboard useful for monitoring the webhook or any other Go process.
To monitor the mutating webhook, see Monitoring the Webhook with Grafana and Prometheus.