When you have a problem, I suggest to use the search feature to find similar problems, often solved.
In particular, there lot of threads already on issues with applets and OpenGL.
The short version is: sign the jars.

I actually currently have it using the P3D engine, is this just a simplified version of OpenGL?  I was seeing posts elsewhere that imply using P3D didn't hit the jar signing requirement.

As an experiment I just threw together a minimal sketch, using the same rendering engine and it had no problem running as an applet (code below).  Are there certain calls that have to be avoided within the applet context?

PFont courier;

int textX, textY;

void setup() {
  size(320, 240, P3D);
 
  textMode(SCREEN);
  courier = loadFont("Courier-12.vlw");
  textFont(courier);
 
  textX = 0;
  textY = 0;
}


void draw() {
  background(12);

  text("This is a test", textX, textY);
 
  textX = (textX + 1) % width;
  textY = (textY + 1) % height;
}

Well, there are lot of sketches in the thread you link to, one of them, near the bottom, using OpenGL, I thought it was the issue.
There was also a bug in a recent version of Processing (1.5?) to read images from the data folder on Windows computers.

It was the final sketch in that thread.  I eventually found the issue, in that it is making http requests from within the applet.  Is this for some reason not possible from within the context of an applet?

You can do http requests without signing the jar... only if the target is the server that provided the applet (same source). Otherwise, more trust is needed (one can send information to an unwanted site) and jars must be signed.