I don't see any rules in the ISSF book about shooting with no shoes, but the RCO would probably pull you up on it for health and safety. In a comp (or any range for that matter) the RCO's word is law.

Get yourself some shooting shoes - they feel very natural, give you a flat base to stand on, support you in kneeling etc. and wear thin socks otherwise you'll blister. I've got some AHG Anschutz shoes that look a million years old in a disgusting poo brown colour, but I shoot better wearing them than anything else. I picked mine up from a member on another forum for £40 about 3 years ago.

I'm not surprised you shoot better with no shoes on. I'll wager you have flat feet/collapsed arches?