Laplace transform using Op-Amps
Is it possible to design a circuit using op-amps such that it's output
  gives the laplace transform of the input signal?


No, it's not possible.  For example, \$F(0)\$ is the integral over all positive time of the input

$$F(0) = \int_0^\infty f(t) e^{-0t}dt = \int_0^\infty f(t) dt$$

So, you would have to wait 'to the end of time' to get the 'initial' output from your box.