I very much doubt the percentage of cowboys who were black, but they were there. They often had cooking or support tasks, but were not accepted by many ranchers or cowboys. Some were taken on by the immigrant Scots English, Irish, and Europeans who settled the northern ranges- Wyoming north into Canada.

The percentage of Hispanics is hard to estimate, but they were numerous in Hawai'i as well as the southwest.

Indians were not accepted until late (the Navajos had an est. 12000 head in the 1830s-1840s and perhaps 50,000 sheep, but these did not enter the market, and were lost when Carleton-Carson et al. rounded the Navajos up in the 1860s), but it must be remembered that the majority of Hispanics from Mexico were mestizos, i. e., mixed white and Indian, and that cattle from Mexico were driven to the northern ranges to feed the rounded-up Indians and fast-growing northern ranches.
Most Texas ranchers used "Anglo" cowboys on drives to the railheads; some of the New Mexico cattlemen were Spanish and of course used local people. People forget that part of Louisiana was suitable for cattle- how many Cajuns?

I dunno how percentage figures can be determined. The figures would also vary through time. I agree, however, that the All white, mostly blond Hollywood image is unbalanced. Also too few Irish, Yorkshire, German, etc. etc. accents in the films- many young immigrants tried their hand in the west.