Assuming you are in the U.S., then it will depend under what rules the race is being run. IIRC, road races under UCI rules won't permit flat bars, but I believe many races in the U.S. (most?) -- road and crit. -- run under different rules, which may permit flat bars (but not bar-ends, aero bars, etc.), or at least don't prohibit them explicitly. You'd have to check with the organizer(s) of any given race for a definitive answer.
Whether you'd want to enter, and other competitors' attitudes to you if you do show up, are entirely different questions of course!