I'd also love this. I push a wildcard cert to my devices, and currently my renewal script uses SCP to push the cert to the right location on certain devices. But this is fragile, software updates sometimes change the cert location or format. An API call to update the cert would be much more robust.
Self-signed certs provide encryption, which is great, but don't have an easy mechanism to provide authentication. It's a better security posture to make sure the browser config doesn't need to be updated to avoid cert warnings. While I could use my Vault instances to run a local CA, that's a big security gap in that X.509 certs are trusted if ANY CA accepts the cert. So if my Vault instance is hacked I'd wind up trusting certs from all over the Internet if I haven't already pinned them. While I'm a small target, I don't have the audits that someone like LetsEncrypt has to ensure there's no drive-by exploits I've fallen victim to. Yes, this implies a more targeted attack, but an API to update certs is an easy ask to close a gap in the security profile.