Search the United States National Archives, at archives.gov/research/arc/, which holds military records going back into the seventeenth century. Go to their archival research catalog and search for "military records from 1872." This search will pull up a plethora of military records dating from and compiled in the year 1872. It will take some time to search through the information from the National Archives. For extra help, call their toll free number to connect with staff. This number is located at the bottom of their website.
Use a service, such as Ancestry.com. If you know whose military records you are searching for, a genealogy website can search particular individual's military records for specific years. These searches will allow you to input the year 1872 for years of service and years of residence in a particular area.
Search through state archives. Each state will have an archive which houses military records pertinent to the activities of that state. If you are aware of which state you would like to research for the year 1872, contact the state archive system. The archive information will be available online through a quick search. A sample search phrase to find a state archive online would be, "New York state archives." These sites will provide contact information for an archivist who can help you locate documents from 1872. This route will be most affective if you know which state you are hoping to obtain military records from.