What is the name of the set defined by nonnegative linear combinations of a set of vectors?
Those special linear combinations are called conical combinations, and the resulting set is called the conical hull of those vectors. The conical hull is always a convex cone.
I believe the name for it is the convex cone of these vectors. In general a convex cone is any set of elements $v,u$ where $av+bu$ belongs to the set again as long as $a,b\geq0$.