Looking for the atmospheric parameters (density for example) and vehicle parameters (CL for example) as well as formulae that were used in the original calculation. Primary sources strongly preferred, as the writeups seen around the interwebs are quite inconsistent.
This is an interesting question and something I had never considered before.. and I think that's kind of the point. The Karman Line has been taken as a fairly universal boundary of space (in the US at least) seemingly without question. Recently, however there has been a fair amount of discussion about redefining the Karman Line. I won't go into discussion on this as it has already been addressed in this question and in a video by Scott Manley.
It appears as though the definition of the Karman line arose during a conference discussion, rather than some published paper. The quote below is from "The edge of space: Revisiting the Karman Line" by Jonathan McDowell.
The ‘von Karman line’ appears to be what mathematicians refer to as a ‘folk theorem’, arising out of a conference discussion but never formally published by him. It was fleshed out in later publications, especially in the influential work of Haley (1963 3) and there is some justification for calling it the ‘von Karman-Haley line’.
von Karman's argument was that the space boundary should be defined where forces due to orbital dynamics exceed aerodynamic forces. A rough order of magnitude argument was used to show that this was at of order 100 km (as opposed to 10 km or 1000 km), but in reality the von Karman criterion defines a line whose altitude varies with position and time (because of variations in atmospheric density due to solar activity) and with the lift coefficient of the spacecraft.
As a result, finding the primary source that you are looking for may not be possible. The closest that I think you will come is Haley's publication which is referenced in the above quote: Space Law and Government.
It's important to note, however, that exact values for atmospheric properties or vehicle parameters aren't generally associated with the definition of the Karman Line. The Karman Line is primarily defined for regulatory convenience, rather than any engineering purposes. In fact, the solar cycle changes the properties of the upper atmosphere so significantly that any accurate definition based on density, etc. would change by 10's of kilometers as solar activity changes.
Another issue is the variety of spacecraft designs; if, as von Karman argued, space was defined where forces due to orbital dynamics exceeded aerodynamic forces, the line would be defined differently each spacecraft. Further, it could even be defined differently for the various orbits and attitudes of the spacecraft.