@John_TYler is correct that if you have a claw with sides rubber-banded together then your force is fixed by width of the claw .
However, if you set it up such that entire claw could move up/down, relative to the rest of the lift, stretching the rubber bands depending on the weight of what it holds, then you could have variable force.
Another way to achieve your goal would be combine rubber bands with something like this:
Similar principle is used in panel carriers.