The best thing I could find on the internet was this apparently forgotten article from 12 years ago.
Mermin has a thorough analysis and traces the phrase to himself in a 1989 Physics Today column  & makes a case that the numerous attributions to Feynman may or may not be mistaken.
 Could feynman have said this? / Mermin, Physics Today, 2004
 What's Wrong with this Pillow? / Mermin, Physics Today, 1989
Some centuries before Mermin, Leibniz in the 17th century was seeking a solution to some of the denominational quarrels that were plaguing his generation by envisioning a calculus ratiocinator that would make it possible for the quarreling parties to "sit down and calculate". A hypothetical science he envisioned was called Mathesis Universalis and included his infinitesimal calculus as a first stage. See also Characteristica Universalis.
As noted, Mermin was probably the first to utter the exact words “Shut up and calculate”. However, the equivalent rallying cry of “Get the numbers out” has its origins some decades earlier.
According to the Nature article History: Shut up and calculate , referring to the role of leading physicist in the war effort (WWII) and how their attitude to their art became largely pragmatic as a result:
This war-forged pragmatism produced enormously impressive research and influenced a generation of leading scientists. Their approach to basic research — and the institutions in which they pursued it — assumed an aura of inevitability. But the approach came with some trade-offs, largely unnoticed at the time. Important questions that resisted the powerful, phenomenological methods tended to get eclipsed. Anything that smacked of 'interpretation', or worse, 'philosophy', began to carry a taint for many scientists who had come through the wartime projects. Conceptual scrutiny of foundations struck many as a luxury. The wartime style was reinforced in the United States by exponentially rising university enrolments after the war. The new classroom realities left little space for informal discussion of philosophy or foundations. The Rad Lab rallying cry of “Get the numbers out” shaded into “Shut up and calculate!”
Here, “Rad Lab” refers to the Radiation Laboratory set up by the American government in 1940 to perform military research.
N. David Mermin (born March 30, 1935, in New Haven, Connecticut, USA) is Horace White Professor of Physics Emeritus at Cornell University
his quote was like this
If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be 'Shut up and calculate!'