How do I pull a backup of an oracle database so I can send it to a vendor
In the Two Day DBA Guide see  see the section called "Performing a Whole Database Backup".  It explains how to use Database Control to do a full backup.  If you have 11g see this version.  Assuming Database Control is working this would probably be the easiest for someone who knows nothing about Oracle.

On the other hand, the database may already have regular backups done on it in which case you can just send one of those.  If it doesn't or the information about them hasn't been retained, then it would seem that the data would not be important enough to convert.