Your family members are found in the US Census of the time (early 1900's), and I'm not listing that information as you may have already researched the documents.However, the Census records show the occupation, which may be the best clue.
I did a name search on Ancestry.com (online subscription to the census records) and did not find "Ray" or "Raymond" Williams in the area.There are a lot of possibilities for reasons why, but they would be only speculation, i.e. "Ray" could have been a nick-name or middle name, the brothers might have owned the shop in Elyria but lived elsewhere, etc.The brothers could have used the cabinet card as advertising.
A good place to look for cards/photographs with photographer name imprinted on the card backing and by location, believe it or not, is eBay.You might be able to do some searching like "Williams Lorain OH* Cabinet Card" (and then check the box that includes the "body of the auction" just underneath the search).Or use "Cabinet Photo*".The "*" behind "Photo" will bring results with "photograph" or other forms of the word."OH*" will bring results with "OH" and "Ohio" and so on.
One option, if you have access to an On-line searchable census, is to look at all the Williams listed in the area in the 1910, 1920, and 1930 census records, and look at their occupations.There are a lot of Williams names listed.If you found them this way (i.e. "Photographer") then you could also look at their families to see if there is any connection.If you don't have access to search a census this way, perhaps a local library might have the capability.