Should I do pushups or other arm strengthening if thin wrists and risk for osteoporosis?

Can't hurt. Bone is stimulated to remodel (create new bone) with weight bearing exercise. Certainly, doing push ups will strengthen bones in the upper body. Carrying dumbbells while walking will help upper and lower bone strength.
Weight bearing exerc. Osteoporosis is a risk for fracture. Usually hips fracture due to more energy absorbed thanthe strength of the bone can permit. If someone has a normal hip density, fracture can occur from a fall from a height of 5 feet. Bones will increase in strength slightly with weight bearing exercise. Working a wrist will increase the strength of the wrist and slightly decrease the risk of wrist fracture, .