[stedaniels]

For NGINX and Apache it's a shell script away. The dehyrated shell client is Let's Encrypt/ACME compatible. It can read from a domains.txt file listing individual certificates per line. There's a hook.sh that you can modify to get dehydrated to deploy the script. All you'd have to do is get whatever script you currently use to provision the custom domains, to add the domains.txt and run dehyrated --cron.

There's plenty of pre-written deployment scripts for it, check the wiki and Google. https://github.com/lukas2511/dehydrated

Failing that... If you're using Apache, it's on its way to natively supporting Let's Encrypt certificates for fully auto HTTPS https://letsencrypt.org/2017/10/17/acme-support-in-apache-ht...

Hope that helps. If you're not comfortable doing it yourself, any dev/sysadmin should be able to cobble it together for a days consultancy.