From the website (http://www.mcnpvised.com/CAD/CAD.html) you can see that this program allows importing 3D geometry from ACIS * .sat file format. SolidWorks support output in this file format also. However, before saving the file, from the Options... button in the dialog box "Save as" you can adjust the version of the ACIS file format to correspond to the supported versions by the MCNP Vised (versions that are supported are 4, 7, 8, 11, 14). For instance, please see attached pictures below.
You can use SuperMC to complete this work. It has powerful neutronics modeling and CAD functions. It can convert a cad file to the input file of popular Monte Carlo programs such as MCNP, TRIPOLI and GEANT4, which is very efficient and accurate. SuperMC also supply Monte Carlo calculation and Visulization.
This is the homepage: http://www.fds.org.cn/en/software/SuperMC.asp