There is no background constraint in maya. I flip in and out of modo just for this feature and I have made several posts in this forum that mention how useful a feature it is. I also placed a request in the official autodesk / maya feature request thread.

There is a very old plugin called "polyshrinkwrap". It is old, really really old, buggy, and way over complicated to use.

Maya is very customizable; However if such a tool / plugin / script has ever been written no one has been willing to share it.

The closest thing you have in maya is Dave's suggestion to try and use ncloth which is complicated and offers very little control or you can make the target surface live and then try to line the camera up as close to normal as possible by eye and then snap the verts (one by one) to the live target surface. Which is time consuming and tedious to say the least.