I've seen people make raised panel doors out of one piece of MDF. I've also seen a professional machine make a shaker door out of one piece. But is it feasible for someone to make one with a handheld router? Would the inset middle area be smooth? If so, what type of bit would be best?
It is perfectly feasible to fab a shaker door from a solid slab of MDF. Routing out the interior flat area is a simple matter of proper jigging and depth control. But you will need to chisel-cut the inside corners and invest a lot of time sanding out the panel area. A router will not leave a 'finish' grade surface when the bit is acting as an end-mill. The best bit to use would be an up-spiral 3-flute bit with 1/2" shank. Bit diameter of 1/2" probably best when balancing strokes necessary to rout down versus difficulty. You'd need to set up two jigs - one as a perimeter jig and another as a platform to ride on the frame while routing down to panel surface.
BUT... That approach is not the only approach to the "MDF Shaker Door" problem.
With MDF as the material, and a router as your only tool (unless you say otherwise), the simplest approach I would suggest would be using two thicknesses of MDF, one for the frames and one for panels. Rout the frame as a unit from the thicker stock, rout a rebate on the back side to accomodate that panel, chisel-clean the corners of these from rounded to square. Then rout the panels out of the thinner stock and glue the panels into the frames.
That will get you there, but is wasteful of the thicker MDF stock (the lost middle parts), and the unsightly joint on the back side.
To do this using the least amount of material, and again with only a router, approach it as a true panel door. Rout the Stiles and rails of the door as individual items from thicker stock, rout a groove into the inner-facing edge to receive the panel (groove size to match panel material). Rout the panels from the thinner MDF sheets. Rout out some splines from the thinner stock, to match depth of groove times 2 and as long as the stiles are wide. Assemble doors with glue, sliding splines in from ends until they contact the panel. Clamp flat.
So, no matter how you go about this you are in for a fair amount of labor. My personal preference would be for the wasteful but simple way, as it asks the least sanding time after assembly.
And in closing, I suggest that you reconsider using MDF at all if that is an option. While it is cheap and in common use it is really not the best stuff to use for a number of reasons. For an inexpensive shaker door, solid pine and pine ply panels would look as good and hold up better over time. And it's doable with just the router, using the last of the options I posted above.