How to make a mesh adjust its form around specific points

Basically i have this object that i want to from its mesh like a sphere/cylinder around specific points. Something like this:

I thought, and tried, to make a mesh in blender and then add bones to it that i could change position of in unity later. But i could not get that to work. The only idea i have right now is to manually change every vertex in the mesh… Any ideas on how to do this would be appreciated.

Rig it inside of blender, And then you can controll the rig from inside of unity

Good luck with your project! Hope I was some help, I know you said you tried it but it seems like you must have did it wrong