Q:

Can I just swap out a .prj file to reproject a shapefile?

I'm an experienced software developer -- but a GIS noob. 
I have a shapefile that I am trying to turn into a SVG map with Kartograph. When I try to run Kartograph I get this error:
ERROR 6: No translation for Lambert_Conformal_Conic to PROJ.4 format is known.

Because I don't want to delve into Kartograph, it seems like I need to reproject the shapefile. The Wikipedia page for shapefiles says that the .prj file describes the projection. 
.prj — projection format; the coordinate system and projection information, a plain text file describing the projection using well-known text format

Can I just swap out the .prj file to reproject the file? 
If so, where can I find a .prj file that will work to reproject the map so that Kartograph can turn it into an SVG?

A:

No, you can't just change the .prj file.  The prj file defines the projection that is used for the geometries, so changing it would simply change how those geometries are displayed to the user.
Consider if you had a shapefile with points in lat/long.  If you swapped the .prj file to redefine the projection to web mercator, all of your points would now be located in the Atlantic Ocean off the coast of Africa, since the coordinates would all be very close to 0,0!
You will have to reproject the shapefile to actually change the geometries to your new projection.  See:
How to re-project shapefiles?
http://support.esri.com/en/knowledgebase/techarticles/detail/21447

