How do I pull a backup of an oracle database so I can send it to a vendor First of all, I know squat about Oracle databases. 

A vendor who is looking at converting our data from an older piece of software with an Oracle DB back-end wants a dump of our database to see what they need to do to bring that data over to our new software which is MS SQL Server based. I know in MS SQL Server, I right-click on the database and say backup, and I can create a backup of the database to a file and FTP that backup to the vendor. How would I do the equivalent in Oracle?

It appears like we might have version 10g of Oracle.
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.