The vendor might want an actual physical backup of your database-- if they do, Leigh's answer is spot on. My guess, however, is that the vendor probably doesn't want a physical backup since that would require that they have an Oracle install on their side on the same operating system with the exact same version of Oracle that you're using. For conversion efforts, it is generally much more effective to get a logical backup of the database using the export utility (classic or DataPump). A logical backup can be imported into an existing database and it doesn't require that they have exactly the same version of Oracle. If all the vendor is going to be doing is extracting the data and loading it into a new database, they aren't going to benefit from having a physical backup over a logical backup.
From the command line on the database server, it's pretty easy to generate a full database export. The command
c:/> exp system/<<password>> full=y file=dumpfile.dmp
creates a single file "dumpfile.dmp" in the current directory that can be sent to the vendor. It would be more efficient to use the DataPump version of the export utility but that requires a bit more work because that requires that you use at least one Oracle directory object. If this is a reasonably small database and/or the time required to generate the export isn't prohibitive, it's probably easier to use the classic export utility.