I believe phagophobia is the fear of eating - or even just swallowing. To get the "in front of people" element included, I'd imagine you'd have to add some element of the Greek word, dimosios, meaning public.
I've just made it up you understand - then that's exactly what psychiatrists do with phobias anyway - but how about dimosiphagophobia. (If you'll forgive the pun, it's a bit of a mouthful!)