How to load .dbf to run select queries?
Toad cannot directly read Oracle data files, no.  You would need to recover the database on a local server.  Once you do that, you could connect to Oracle database on the local server from Toad and run whatever queries you'd like.

Assuming that what you've been given is a consistent cold backup of your database, you'd need, at a minimum, exactly the same version of Oracle (patchsets included) on exactly the same operating system that the third party vendor uses.  Assuming that the vendor is vaguely competent about applying patches, that will likely require that you have a My Oracle Support account which would require that your organization has appropriate Oracle licenses.  How you do that will depend on the operating system your Oracle server will be using.

Since it sounds like you are not a DBA and you're not particularly familiar with Oracle, assuming there is a reason that you can't simply access the existing database to run whatever queries you need to run, I would tend to suspect that you would be much better off requesting an export of the database (or potentially the tables you're interested in if you really care about a subset of the data) rather than a backup.  Toad can't read an export directly either but it is a lot easier to import and export file into a database you create on your local server without worrying about matching the version and operating system that the vendor uses or going through the effort of doing a proper database restore.