I am looking for a program, user script, API, or web tool that can indicate, for a given researcher, what percentage of their publication is open access.
A free option would be ImpactStory. Here is an example of a person with 100% OA articles.
Other than that, commercial databases such as Web of Science or Scopus contain such data.
This screenshot shows an example of an Author Profile search whereby in the left-handside menu you see the option to refine your search to 'Open Access' publications. One can also download these data as CSV files: