Is there an original source for this photograph?
Fortunately, there is. This photo is originated from an official source.
Frank Sinatra was honored with an official $1,500 book limited with 1,000 copies.

The book's description is:
The official luxury book to commemorate the Frank Sinatra centenary, limited to just 1000 copies, in a deluxe clam-shell box, accompanied by a previously unpublished photograph, taken and authenticated by Nancy Sinatra Sr. Each book contains a numbered certificate of authenticity, signed by Sinatra's children - Nancy Sinatra, Frank Sinatra Jnr., and Tina Sinatra.
The book is signed by the whole family:
This spectacular 400-page book - which comes in luxury clam-shell case - is signed by Nancy Sinatra, Frank Sinatra Jnr., and Tina Sinatra and includes a rare and unpublished signed print of a photograph of Frank Sinatra taken by Nancy Sinatra Snr (see
image below, middle of the row).
At the end of the page, many preview pictures are found along with the picture you're skeptical about.

Additionally, the book is available officially on Amazon along with the picture also:

Who shot the image?
officialnj350.com says in a PDF document that Michael Ochs Archives shot the image and it is published on Getty Images.

Is this photograph of Frank Sinatra as a 10 year old?
Not necessarily. It is Frank Sinatra, but it is reported that the picture is undated. He could be 8, 10 or 12, etc... The article says:
Behold: It’s Frank Freakin’ Sinatra as a young boy. The photo is undated, however Frank looks to be about 10-years-old.