Compared to what's available for service members in earlier conflicts, very little documentation about WWI service members and post-WWI service members is online. NARA's page Request Your Military Service Records cautions:
Please note that NPRC holds historical Military Personnel Records of
nearly 100 million veterans. The vast majority of these records are
paper-based and not available online.
Start with Home Sources
Start with home sources to see if you can locate something with his service number, as that will help you distinguish your veteran from other people with similar names, and look for what unit he served with, because that will narrow down both the time he might have been in Germany, and the area in which he served. Another critical piece of information you'll need is the date of your relative's separation from the military -- this date determines what restrictions there will be to access his service file, and may give clues about what happened to his file.
Download a copy of Standard Form 180, so you will know what information you will need if you want to request his records. Only a veteran and the next-of-kin can order via the online system; you may have to order his record via mail. But you'll want this form for your records, no matter what.
The next-of-kin is defined as any of the following: the un-remarried widow or widower, son, daughter, father, mother, brother or sister of the deceased veteran.
If any of your grand-uncle's next-of-kin are still living, have them submit the request instead -- you will get more information that way.
When did your relative separate from the military?
NARA's Veterans FAQ explains the rolling window that determines whether records are archival or not. Under 1. What is the difference between Archival and Federal (non-archival) Records it says, in part:
In 2004, NARA together with the Department of Defense (DOD) developed a schedule making the Official Military Personnel Files (OMPF) permanent records of the United States. This schedule mandates the legal transfer of these files from DOD ownership to NARA ownership 62 years after the service member's separation from the military. Separation from service is defined as discharge, retirement or death in service based on a rolling date. For example, if today's date is January 1, 2018, then the discharge, retirement or date of death must be January 1, 1956 or before to be considered archival. Archival records are no longer the property of the agencies that created them, in this case the Military Service Departments, but are records of the National Archives, open to the general public. See Archival Records to access these records.
If your great-uncle separated after that, see Access to Official Military Personnel Files (OMPF)for the General Public. (This article contains the definition of the next of kin quoted above.
Learn More about the Records before you spend money
Take a step back, and learn more about what other records you might be able to discover. You don't want to leap in and search for the records themselves, but for research guides, so you can put any of your finds in context.
At the National Archives website, start with:
If video learning appeals to you, Michael Strauss' webinar The Great War: Researching Your World War I Ancestors gives a good overview of what records are available. It is available as part of Ancestry Academy, and at Legacy Family Tree Webinars for purchase as a digital download or as part of a subscription (1-month subscriptions are available for less than $10). If you are a member of a genealogical society that has a webinar series, check their webinar library; the webinar has been presented at SCGS and is in their library for members to view, and it may be available through other genealogical societies as well.
The Fire of 1973
In other circumstances, ordering your relative's service records from the NRPC in St. Louis might have been a simple way to answer your question, but the estimated record loss for WWI Army service records is 80%. Ask St. Louis if they have a record, but it might not survive. In that case, you'll have to look for the answer in other records. Since we don't have your relative's date of separation from the military, it isn't clear whether his OMPF will have been involved in the fire or not. But even if it survived, the information you want may be in other records.
We don't know if your grand-uncle ever asked about bringing these children and their mother back to the United States, but at that time, service members needed permission to get married. If any official notice was taken of his situation, there might be evidence of it in unit records that were not stored in the areas of the NPRC involved in the fire.
See NARA's guide Military and Veterans Research and Resources at The National Archives and the linked article 20th-Century Veterans' Service Records, and the section Military Records Holdings in St. Louis, Missouri on the page Access to Military Service and Pension Records.
My guess would be that there might be correspondence in
165.4.12 Records of the Office of the Executive for Personnel and Administration--Personnel Branch
which is made up of:
Textual Records: General correspondence, 1941-45. Subject correspondence, minutes, and reports, 1942-44. Daily activity reports, 1943-44
(copied from the overview of Records of the War Department General and Special Staffs)
but that is only a guess. I am just starting out in research in military records, and this is not an area where I have a lot of experience.
This is a case where it might pay off to hire a researcher who is experienced in 20th-century military research to hunt out information for you. They will have a better idea of what records to search than hobbyists like me who only dip into military research from time to time. But learning more about the records yourself will allow you to understand the scope of the question, and help you to communicate exactly what information you hope to find.
Above all, keep in mind that if your grand-uncle kept this relationship secret from his superior officers, the most you can hope for from his OMPF is information that will give you times and places that he served, thus narrowing the search field for other records.