Sit so that the weeds to be pulled are in front of you. Then, instead of bending over at the waist to get a hold of the weed, try moving your trunk as one unit, keeping your spine long as you go. There are a few keys to doing this successfully. First, a folding action at the front hip joint will enable you to move your torso as a unit. Second, to get your trunk moving, start by lifting the back hip off the ground just a bit. This will provide a push forward for the rest of the body.
Also, try to keep the angle of your arms the same throughout the movement. This will be especially useful when you pull the weed out of the ground. For now, find a comfortable position for your shoulders as well as an angle of the elbows you can sustain, and don't vary them as you move to grasp the weed.