Assuming you mean this:
http://vgcollect.freehosting.net/myvectrex.htm#FullMulti
I think the info in that PDF is mainly aimed at others
designing multicarts from scratch. You could try Emailing the
author for more info, but it's a bit of a long shot with a
project that old.
If you aren't up to designing your own one, I'm not aware of
any other DIY projects. If you just want one game, you might
be able to use an EPROM, but then you'll need to buy a
programmer (unless you have, or know someone with, one
already), this will likely blow your $20 budget from the
start.
Here is a project for sacrificing a standard Vectrex cartridge
to make an EPROM cart. (eprom.txt):
http://web.archive.org/web/20051216165552/http://www.vectrex.c
om/Mirrors/FTP_Mirror/text/
**There was a post here in 2011 regarding access to that FTP
site. Now both the links that came up there are dead, so I'm
stuck with the Wayback Machine archive above. Is there any way
left to get to the FTP site?**
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