As Mencken has so amply documented, Americans are well known for their adroitness in coining a catchy and appropriate phrase, yet they seem to have been a little slow in connecting the elasticity of caoutchouc with the craning of the overcurious.
Rubber, as a contraction of India rubber, first appeared in print at least as early as 1788, but it was over a century later, in 1896, before rubberneck did so.
To be sure, it was probably known in the spoken language a little sooner, but Americans also have the practice of publishing their catchy phrases soon after they are coined, so very little additional age can be assigned this colorful term for a snooper.