The bodily functions do not require the brain when life support is turned on.
Ignoring of course any modern scientific medical treatment that may help sustain an organism for some time while their body has stopped working naturally, it cannot be ignored that every function in your body functions due to your brain actively instructing it to do so.
Your question was simple, "what keeps all the organs functioning?" - you did not add any other specifics to it.
It's an indisputable fact that your organs get their instruction to do what they do from your brain, shut your brain down - NOT adding any life support - and your organs will shut down as well.
Again, "what keeps all of the organs functioning?" - the brain
"I smoke pot. If this bothers anyone, I suggest you look around at the world in which we live and shut your mouth."--Bill Hicks
"I never knew there was another option other than to question everything"--Noam Chomsky