We have an old plum tree in our yard that has grown bent over due to pine trees right next to it. Last year some small trees sprouted up underneath this tree, so I thought about transplanting them throughout the yard. I discovered when I went to dig one up that they are actually growing right off the existing root system. Is there a way to transplant them without them dying?

Read this article about tree "suckers." It should be helpful to you: Tree Sucker Removal And Tree Sucker Control (http://www.gardeningknowhow.com/ornamental/trees/tgen/tree-sucker-removal-and-tree-sucker-control.htm)

So there is no way to save the "suckers" then?

So there is no way to save the "suckers" then?
No. They will not grow to be plum trees.

So there is no way to save the "suckers" then?

No, they don't have their own root system.