well i dont know the shoutbox, but i assume its calling also php files..
so i think what you basically should do (I may be wrong...)
1. go through ALL the shoutbox templates and make the urls absolute ( that should work as you basically dont do anything else than a base href but for shoutbox only)
2. for all the urls generated by the php files ( I dont know if there are any), do the same. (add the http://www.yoursite.com/forum
... in front)
3. test it.. if it works replace the http://www.yoursite.com/forum
with the vbulletin variable: $vboptions[bburl]
in the templates and $vbulletin->options['bburl']
in the php files.. so if you migrate to another server.. you shouldn't have problems..
but as I said.. i dunno if it works.. because i never used any shoutbox.