Besides policy files and the signing of applets, a couple of possiblities/suggestions:
Applets seem to balk at the creation of files.
Sometimes you can get around this by avoiding the File constructors and using URLs instead.
URL url = getClass().getResource(imagePath);
BufferedImage image = ImageIO.read(url);
Another possibility can be to load the images in a class that is declared outside the body of the applet.