Were oppressed people able to make progress toward equality during the
people of color (African Americans and Native Americans), white women, and immigrants - and the struggles they faced in their attempts to achieve full citizenship in the U.S.
I would say that one of the critical elements of Progressivism was the idea of expanding voices into the American social and political expressions of reality. I think an argument could be said that groups which were silenced received greater empowerment through the Progressive Era. For example, due to the Muckrakers, issues of consumer safety and worker's rights were addressed, concepts which the Industrialists ensured were not heard. Booker T. Washington, Ida B. Wells, and W.E.B. DuBois helped to articulate the condition of African- Americans and how their collective voice should be heard. Women fought to achieve social and political equality, represented in the passage of the 19th Amendment. Certainly, much, much more was needed, but the Progressive Era marked one of the first steps on the long march to equality.
It is also important to note that, while gaining the right to vote with the 19th amendment, social attitudes towards women in the workplace (outside of World War I factories), and towards their roles as wives and mothers remained conservative and traditional.
The conquest of Native Americans was completed by the end of the 1800's during the Progressive Era, and the Dawes Severalty Act confined them to reservation status and boarding schools. It was largely the opposite of progress for them.
Immigrants came in large numbers from 1880 - 1920, and xenophobic (anti-immigrant) feelings led to anti-immigration laws like the Chinese Exclusion Act, and later quota systems.
The answer for each of these is a bit different.
White women made the most progress towards equality during this time. In fact, women were the driving forces behind some of the major reforms of the time, including Prohibition and the settlement house movement. In addition, of course, women won the right to vote at the end of the Progressive Era.
African Americans got very little tangible benefit from this era. However, the NAACP started up during this time.
Native Americans essentially got nothing.
Immigrants were never deprived of citizenship rights like the other groups you mention. However, this time was sort of ambiguous for them. Many of the reforms helped them, but many of them (especially Prohibition) tried to force them to conform to a "native" American, middle-class idea of how people should be.
In history of the USA the period between about 1890 to 1917 is often described as progressive era because of the many economic, political and social reforms brought about in the life of people by some people who called themselves progressive and who worked to bring about changes directed at reducing poverty, improve working condition of the poor, regulate big business, end corruption, and many other similar goals.
During this period American Federation of Labour (AFL), now called American Federation of Labour-Congress of Industrial Organizations (AFL-CIO), was formed. The movement for women suffrage became very strong during this period, which led to women gaining right to vote in local elections in some states.
Many laws were passed by local and state governments to help the poor. These laws promoted causes and practices such as improvement in housing and related amenities, employee safety, industrial disputes settlement, and education. In general the movement led to greater government regulation of business and industry.
The 17th amendment to the US constitution adopted in 1913 that provided for direct election by the US people of senators was also a part of the political reforms of progressive era.
However there does not appear to be much concern during the progressive era towards improving the condition of minority groups such as African Americans and native Americans.