How To Make Servo Brackets Like Those On Lynxmotion Robots?

They look like bent metal. The easiest way would be to get some sheet aluminum, preferably from a scrap yard. make a small wood block the size of the interior space and bend the aluminum around it.

If you are going to do a number of these, think about making a simple brake to make the bend uniform. This can be made from a sjort piece of angle iron and a couple of hinges. A handle on the side will give you some leverage. Here is a link:

Make a little jig to rotate the bent bracket against a grinding wheel to give you the rounded corners.

Do a production run since setting up the jigs will take a bunch of time. Doing one may take five minutes, but the second one will be 15 seconds.