Does this voltage to current amplifier setup work?
It would work if you fixed it: -


The red box is connecting the power supply to ground correctly
The purple box is fixing the supply polarity
The blue box is a warning to take care you have the right power rails on the op-amp


Does this mean the load can be a BJT controlled circuit with its own
voltage supply without affecting the current through R2?

Not with this circuit but if you want a ground referenced load then using a PNP transistor and R2 connected to the positive supply can achieve constant current through the load - see circuit 4 in this answer to see what I mean. Also this answer and this answer and this answer for the same example and some theory.