I can use the API - https://<domain name>/rest/api/latest/issue/<issue id> - to get the full JSON payload for a Jira issue.
Is there an API I can use to only get the text on the documentation tab for a Jira issue?
You can use the fields query parameter to select which fields should be returned by the API. If you add "&fields=key,documentationfield1,documentationfield2", the API returns just those fields.
You can find additional info on the fields parameter in the docs.