It takes five points to define an arbitrary ellipse in a plane, not knowing a priori where the center is. The five points, or really any three of them, also define the plane in three dimensions. (You never said the number of dimensions in your problem, so I am assuming three.) You will then know where the foci of the orbit are, but you won't know which one is where the body is, with no information about time.
It takes three points to define an ellipse if you know where a focus is.
In both cases, it is possible to pick points that give you no new information, so not all sets of five or three points on an ellipse will give you a unique solution. However those points are of measure zero out of all possible points, so if you pick points randomly as opposed to maliciously, five or three will be enough.
In neither case will you really define the "orbit", since you don't know anything about time. E.g. what the orbit period is, unless you know the mass of the central body, which you made no mention of, and even if you know that, you don't know when the object is at any given position in the orbit. So you are left with one degree of freedom that cannot be determined, no matter how many positions you have.