Thank you for posting your request on History Hub!
What a wonderful gesture to try to track down the family of Henry K. Richey to return this book. Doing this level of research can take a great deal of time and patience. Luckily, quite a few people have started creating family trees online and tracking down family members has become much easier in recent years. It sounds like you have a little information on him to start your research. You could search the American Expeditionary Forces in our catalog, although this might not lead you to family members.
Probably the best place to start is with census records. I did a quick search and there are quite a few Henry K. Richey’s listed in the census but if you know can estimate his date of birth and the area he was living during the census that's a great place to start. The 1940 census is available on our website here. Earlier census are available through Ancestry, which is a subscription based site but is free through most public libraries or at home with your library card. They're also available through FamilySearch which is free, but you'll need to create an account.
Once you find some records of his you may be able to follow his family through vital records, such as birth, marriage and death records which are recorded at the city or county level. These are often available on Ancestry and FamilySearch as well, but you may need to contact local resources if those records have not been scanned. At this point in your research, you may be able to find a good match for him on a family tree or track down a living relative through the resources listed above.
We hope this information is helpful. Best of luck on your research!