You'll actually have to get into the new vSphere 6.0 vCloud Suite SDK for REST (VMware vCloud Suite SDK for REST for vSphere 6.0 - VMware Developer Center).
Prior to vSphere 6.0, you probably need to look at the PowerCLI commands. There's an internal API that's used against the inventory service in 5.x. I did some packet captures can you can probably replicate the call (mix of xpath + SOAP). But didn't seem worth the effort.
Which internal API can we use to get the vmware tags and tag category?
I didn't spend enough time on it, just did a few SSL packet captures. It's basically the SOAP API, just they seem to embed some XPATH operations for the tag management inside the content. I honestly recommend just not using vSphere Tags, unless you're really bumping into functionality instead of metadata tagging. You can get the same result with custom values for the most part when looking to just do tag lookups.
It's been something we've raised as an issue several times to the PMs, but it hasn't gotten any traction outside of the vSphere 6.0 REST API (so only PowerCLI support pre-6.0).
