How to wipe an old white MacBook?
The short answer is...

Using OS X boot media or your recovery partition (if available; Command+R), boot into it like you are going to do an install.  However, when you get to the first screen, launch Disk Utility and do a "Secure Erase" of your hard drive.

Once that is done, you can stop there and have a white MB with no OS or you can continue and install an OS for the next person who becomes the new owner.

Can your white MB support Internet Recovery?

If it is the 2010 model (their last one before resurrecting the line last year) and you had Lion installed with the EFI Firmware update, you can use Internet Recovery.  https://support.apple.com/en-us/HT202313

There's no harm in checking...just press Command+Option+R during boot.  If you get to Internet Recovery, then you have it.  If not, then chances are, your MB doesn't support it.