Can food swallowed when a person is upside down reach the stomach?
The eminent @JanDvorak has basically provided the full answer, but yes, it is possible.  You can easily try this yourself, just lean off a bed and swallow something (small) and see what happens.  Peristalsis in the esophagus is responsible for moving food down into the stomach.  The big proof of this is that astronauts can eat when in space, when weightless.  Their food is moved into the stomach just fine.

It's easier to do it upright (peristalsis + gravity) as opposed to upside down (peristalsis - gravity) but definitely quite possible.