Is it possible to simulate a quadrupedal robot using Moveit? If not what is the best possible way to control the robot through ros?

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In addition to what @v4hn already suggested, you might want to take a look at "ANYmal at the ARGOS Challenge: Tools and Experiences from the Autonomous Inspection of Oil & Gas Sites with a Legged Robot", a presentation by the ANYmal team (from ETHZ ASL). Video here, presentation: slides.

From the ROSCon16 program:

At the ARGOS Challenge, we participate with our legged robot ANYmal in the autonomous inspection of oil & gas sites. In this presentation, we share our experience with using ROS in our work and introduce our tools, many of which are available open-source. ANYmal is an electrically driven quadrupedal robot, capable of dynamic running, climbing over obstacles, and scaling stairs. It uses onboard laser range sensors to precisely localize and map its surrounding, enabling the autonomous navigation in difficult environments. For ARGOS, ANYmal carries several inspection sensors and uses computer vision to monitor the state of the site.

MoveIt gives you tools to work with your robot kinematic structure, visualize, check for collisions, plan trajectories, etc.

You can setup walking robots with Moveit and I know of at least one ongoing project that builds on MoveIt's infrastructure for bipedal walking.

However, of course this will not "magically" make your robot walk.