This is almost certainly false. The following animation shows two convex shapes (with outlines shown in red and green) whose Minkowski sum is a disk of radius 3 (with outline shown in blue). The green shape is an ellipse with major and minor radii 1 and 1/2, which uniquely determines the red shape.

I do not have a proof that the red shape is convex, but it shouldn't be too hard to check.
Incidentally, here is the Mathematica code I used to produce this animation:
MyPlot = ParametricPlot[{3*{Cos[t], Sin[t]},
With[{u = ArcTan[-Sin[t], Cos[t]/2]},
3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}]},
{t, 0, 2 Pi}];
myframes =
Table[With[{u = ArcTan[-Sin[t], Cos[t]/2]},
With[{pt = 3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}},
Show[MyPlot,
ParametricPlot[pt + {Cos[r], Sin[r]/2}, {r, 0, 2 Pi},
PlotStyle -> Darker[Green]],
Graphics[{PointSize[Large], Point[pt]}]]]], {t, 0, 2 Pi - Pi/20,
Pi/20}]; ListAnimate[myframes]
Edit: Here is a simpler solution using two congruent shapes. The boundary of each shape is the union of two circular arcs, each of which is congruent to 1/4 of the blue circle.
