In an effort to help you, I ran a Google search for the exact phrase "Vonnie Slaughter". I found the newspaper archive almost useless. Then I read two queries you posted on a genealogy website. They linked Vonnie to Sullivan County, TN.
I checked the Social Security Death Index (SSDI)http://ssdi.rootsweb.com
Using the Advanced Search, I checked for decedents named SLAUGHTER whose last address (known to the Social Security Administration---SSA) was in Sullivan County, Tennessee.
On that list was a Lonnie SLAUGHTER (1901-1967), last address in Bluff City. I do not claim this person is the birth mother of your husband. In fact, Lonnie could be a nickname for the male forename Alonzo. If you wish you can compare the decedent's birthyear with that implied for Vonnie (on your husband's birth certificate).
If a person did not have a Social Security card or his death was not reported to the SSA, he would not be on the SSDI. If your husband wishes to apply for a copy of Vonnie's death certificate (if any) after guessing at the approximate year of death and what state it occurred in, obtain the address of the vital records office from this online list :http://www.cdc.gov/nchs/howto/w2w/w2welcom.htm
He can prove he is a descendant of the decedent Vonnie by submitting a copy of his birth certificate (showing her name) and complying with whatever regulations are on the application form. Some states want to see photo i.d. for the applicant.
Your husband might be interested in seeing the federal census of 1950, closed to public access for 72 years.
If he needs proof of age for a certain purpose, such as applying for Social Security or a passport, he might be able to get Age Search information now from the federal government from a census which has not yet been made public, such as the 1950 census. The information might be limited to his own entry plus the name of the head of household. The U.S. Census Bureau has an office in Jeffersonville, IN which will provide an application form with additional information. If you cannot find it on the Internet, call 812-218-3046.