The address is not the location of the farm, but is the post office box. The post office in Lewisville was located n the what was 1st Street, which is now Arkansas Highway 82. That address format was not uncommon before the modern zip-code system for P.O. Boxes was adopted in the 1970s. So he probably lived on an outlying farm and gave his mailing address on the draft card. Even today, some rural places in the United States do not have individual household mail delivery, but residents come to the post office to pick up their mail.
In the 1940 census, he is listed with his family of 7 children. The eldest child is listed as Clarence (born ~1917). I was able to find a reference to an obituary for Clarence A Brown, born in 1918 in Lewisville, Arkansas and died in June 2000 in Oakland, California. This is likely the first son. If you are able to get the full contents of the obituary, it will likely have information on living relatives.
The obituary was printed in the Oakland Tribune on June 20th, 2000. You'll need to find someone that is able to do a lookup (probably at a public library in Oakland) to get the contents of the obituary. I'll see what else I can find for the other children.
You might also be able to get some help from /r/genealogy. Usually they trace going from the other direction, but they could clue you in to some resources.
Edit: Found more info. The son, Clarence, also served in WWII. His full name is Clarence Alfonsa Brown.
There is both an Alameda city and an Alameda County in California (the city is in the county). The city of Oakland, where another poster said this man's son died, is also in Alameda County.
I live in Alameda County, and just for kicks I called the one big cemetery that has the vast majority of burials in the county. They have no record of him. Nor is he in the national database as having been buried at a military cemetery. But if you find information he was of a particular religion, there are Catholic and Jewish cemeteries here I could also check for you. I know you're really looking for his kids and not his grave, but that's what I've found.
Death records are handled at the county level. A death certificate would not list any children, though, only parents and spouse. It would also list the cemetery.
You can order a non-certified copy of the death certificate here:
http://www.cdph.ca.gov/certlic/birthdeathmar/Pages/CertifiedCopiesofBirthDeathRecords.aspx
I could be way off, but I was messing around on FamilySearch also with this. Is it possible that this is the dog tag for Joseph Brown, the son of the one you are looking at. Since he is single, could the next of kin on the tag could be his mother? He was born in 1919 making him in his early to mid 20's during the war.
At age 11 (1930 census) he resided at Lewisville, Lafayette, Arkansas, United States.
Edit: Also, if this family is African American. He most likely did not fill an active combat role (still segregation). This means he likely survived the war to return home.
Ok, I think I've found him. It is his son. Joseph Martin Brown. Goes by Martin in a marriage event to Allene Hammond on Dec 26, 1949 in Cullman, Alabama, United States. Born ~1919 or 1922. Employed by an airline and time of marriage.
I'm sorry, Reddit TOS forbid the posting of personal information. This question has been removed.
Putting together some of the info. so far, if the family is African American and they moved here to Alameda County after the war and the subject is Joseph Brown Jr., this might be his father's grave at Evergreen Cemetery in Oakland:
I called the cemetery, and they also have othe Joseph Browns who were buried in 1977, 1982 and 1984. One of these is buried in the "Veterans' Alcove". It's a common enough name so it might not be our guy, but we could have the death date wrong.
I'll check for obits of his other kids in a bit.