We use an ultrasonic standing wave to simultaneously trap and deform thousands of soft lipid vesicles immersed in a liquid solution. In our device, acoustic radiation stresses comparable in magnitude to those generated in optical stretching devices are achieved over a spatial extent of more than ten acoustic wavelengths. We solve the acoustic scattering problem in the long-wavelength limit to obtain the radiation stress. The result is then combined with thin-shell elasticity theory to form expressions that relate the deformed geometry to the applied acoustic field intensity. Using observation of the deformed geometry and this model, we rapidly extract mechanical properties, such as the membrane Young's modulus, from populations of lipid vesicles.