Yes, I was thinking that you might have a plugin already installed, but which doesn't work with vb4. That might be easier since it might only need small changes.
I was about to say that it couldn't be done with template edits and that you probably need a plugin, but that might be wrong: you might be able to edit the postbit_attachment template and do something like:
Code:
<vb:if condition="$attachment['attachmentextension'] == 'pgn'">
// applet code from link above here
<APPLET archive="Viewer-Deluxe.jar" code="ChessBoard.class"
codebase="/bin" width="631" height="560" mayscript="true">
<PARAM name=PgnGameFile value="attachment.php?{vb:raw
session.sessionurl}attachmentid={vb:raw attachment.attachmentid}&d={vb:raw attachment.dateline}">
<PARAM name=LightSquares value=F3DCC2>
<PARAM name=DarkSquares value=DDA37B>
<PARAM name=Background value=CCCCCC>
<PARAM name=ImagesFolder value=images>
<PARAM name=PuzzleMode value="off">
Your browser is completely ignoring the <APPLET> tag!
</APPLET>
<vb:else />
// existing template code goes here
</vb:if>
Note that you need to keep the existing template code and make sure it ends up where the comment is. Also, you need to set the codebase= to point to where your Viewer-Deluxe.jar is (unless you actually do have it in a /bin directory).
BTW, I haven't tried it so I don't know if this works, but it's the best I can do without actually doing it myself.