I recently saw the boards of a flat-Earther on Pinterest. I believe that the notion that the Earth is flat is complete idiocy. I had a feeling that something would be wrong with gravity if the earth was flat, but I wasn't sure what. So that brings me to the question, could a disc, given enough mass, have a gravitational pull, and if so, how would it act?
Yes, anything that has mass will exhibit a gravitational field according to Newton's law of universal gravitation, and the strength of the field will be weaker at far distances from the object, but stronger closer to the object, roughly speaking. More precisely, the field will weaken "as $1$ over the distance from the center of mass of the body, squared." So yes, a flat earth would have a gravitational attraction, but things would look quite different than the way they look on our round earth. If you were near the edge of the disk, you would appear to be standing almost perfectly sideways since the gravitational field points towards the center of the disk, not straight into the disk, and walking towards the edge would be like climbing a mountain becoming steeper and steeper. This is neglecting the fact that the disk would collapse under its own gravitational pull, but is something to think about.
Note that most flat-earthers deny the existence of gravity as I described it here. They instead assert that the flat earth accelerates uniformly upwards at $9.8$ m/s$^2$.
Here is a wonderful VSauce video on YouTube that discusses the flat earth. There is an animation showing what gravity would look like on the disk-shaped planet.