Air suspension unit for relatively low mass
I don't think (like the other answers) a suspension fork will do it.  There's another reason as well - suspension forks only isolate vibrations in one linear dimension, as you go over bumps, as well as up and down you will accelerate and decelerate, and pitch up and down.  So here are some other thoughts.  We might be able to offer more help if you give some idea of what the object is.

It's not unknown to use an underinflated inner tube as a shock absorber for light loads - but that's generally a flat object on a flat surface.  Providing suspension for a light object can be done by hanging it in a frame using bungee cords (shock cords).  The rubber provides some damping so it's better than just springs.  Might be hard to fit between you and the bars - but then if I was doing 30km/h over rough ground I wouldn't want a set of forks pointing at my chest as you've shown.  Either on a modified rack behind or in a modified front basket would be better.

Whatever means you use, for best performance you need to tune the system - with some means of sensing the vibration (smartphone app these days?) and adjusting either the thickness of the cords or the mass.