If the Earth is flat, wouldn't gravity pull you towards the center of mass of the Earth, which, if you are near the edge, would not pull you perpendicular to the ground?
In some other post, I pointed out that the FE'ers really have four options for "what is gravity":
1) Infinite disk earth with gravity.
2) Infinite disk earth with Universal Acceleration.
3) Finite disk earth with gravity.
4) Finite disk earth with Universal Acceleration.
The folks here at TFES.org seem mostly to be agreed on (2). Personally, I think they'd be better off with (1).
The problem you describe would indeed be an enormous problem with (3) - so as far as I can tell, the "finite-flat-earth" people either haven't thought about it - or are going with (4).
Option (2) has problems though. For UA to work would require an infinite energy source...and would either violate the law of conservation of momentum - or would require the earth to be sloughing off material from the underside at some prodigious rate as it accelerates.
So if *I* was an FE'er - I'd be going with option (1)...but all four approaches have problems with the variability of gravity over the Earth's surface (greater at poles than equator and less on top of tall mountains)...and falls apart when you consider tides.
A VERY few FE'ers claim that effect that we call "gravity" is caused by air pressure pushing down on things - but even their fellow FE'ers find that funny. After all, if you put something under a bell jar and pump all of the air out, the object doesn't float. So that's a bust.