Preamble
Suppose I have a finite collection of points in $\mathbb{R}^n$, and I compute a Delaunay tesselation on them. In this $n$-dimensional space the straight-line edges are non-intersecting line segments. I am under the impression that there always exists a map $T: \mathbb{R}^n \mapsto \mathbb{R}^2$ that would arrange the points such that the resulting embedding of the Delaunay tesselation on these points would be planar.
Examples
This is the projected Delaunay triangulation for Ronald Fisher's Iris data, which is a collection of points in $\mathbb{R}^4$. The red scatter shows points of intersection for the lines representing edges of the Delaunay tesselation. This visually makes clear that the transform I chose does not produce a planar embedding. The transform was a linear transform via principal component analysis, which I had no expectation would work for this purpose but would allow me to develop such a figure.
As another illustration, here is an embedding of the same Delaunay tesselation using tSNE.
Question
For a given collection of points, is there an algorithm for constructing map that will provide a planar embedding of the corresponding Delaunay tesselation?
Your impression is not correct. Delaunay tesselations are equivalent to polytopes inscribed in the sphere, and so you are asking whether the 1-skeleton of an inscribed polytope is always a planar graph. That this is false follows, for example, from the work of Gonska and Ziegler:
Gonska, Bernd; Ziegler, Günter M., Inscribable stacked polytopes, Adv. Geom. 13, No. 4, 723-740 (2013). ZBL1288.52007.