of course now that someone has pointed out that you can build a 1-DOF quad, replicating that idea would only require 4 cheap $9 servos and for a prototype you would make a long cable to the electronics. You could literally use hot glue to stick four sevos into a brick and srew sticks to the servo horns and have a working quadruped in 15 minutes.
Lots of credit has to gop t=o the person who figured out this could do something interresting.
OK, yopu could build it but could you make it walk? It would not be a natural gate at all. Legs motion has two phases. Stance phase when the foot is in contact with the ground and a swing has when the foot is in the air, moving forward. The question is how to lift a leg off the ground. It seems they do that my using two diagonal legs to lift the robot body and the other two legs with it.
This is relly interresting as a beginner project. We usually bud a diferential drive mobile platform because it is the easiest thing to build. But this could be done, at least the mechanical part, for under $20 in an hour or so.