Welcome aboard from ol' Wyo.
Thanks for showing us your Mom's Chiefs Special. Your Mom has good taste. As you probably noticed on the inside of those very nice stocks, they were made by Steve Herrett's Stocks in Twin Falls, Idaho.
The serial number is in the first Chiefs Special serial series, which ended in 1969 at 786544. Between that a few lower than yours in the database I'd guess summer of '69 is a safe bet for when Smith & Wesson shipped yours.
You can also see the five-digit assembly number on the lower left side of the grip frame (with the stocks off). The third location it was stamped is inside the side plate. The numbers were used in the factory to keep the parts together through the assembly process. They serve today to verify the parts are original to the gun--most guns are still intact as they left the factory so I do not recommend removing the side plate just to see the number.