Yes, it is possible to trap a single light ray in a polygon.

Mitchell, Zachary, Gregory Simon, and Xueying Zhao. "Trapping light rays aperiodically with mirrors." Involve, a Journal of Mathematics 5.1 (2012): 9-14. (Journal link.)

Abstract. We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically.

To obtain a polygon, one would have to connect their disjoint segments into a path, but I think this would not be difficult.

But it is not possible to trap light rays from a continuum of directions:

Dawson, RJ MacG, B. E. McDonald, J. Mycielski, and L. Pachter. "Light Traps." (1996). (PDF download.)

Update. I answered too quickly. The construction I describe traps a ray whose source is inside.

Mitchell, Zachary, Gregory Simon, and Xueying Zhao. "Trapping light rays aperiodically with mirrors." Involve, a Journal of Mathematics 5.1 (2012): 9-14. (Journal link.)

Abstract. We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically.

To obtain a polygon, one would have to connect their disjoint segments into a path, but I think this would not be difficult. Update. Apologies. Now that I found their construction, which mimics an irrational sloped billiard path reflecting inside a square, it is not immediately evident how to inject the ray from outside the construction...

Incidentally, it is not possible to trap light rays from a continuum of directions, even with curved mirrors:

Dawson, RJ MacG, B. E. McDonald, J. Mycielski, and L. Pachter. "Light Traps." (1996). (PDF download.)

Yes, it is possible to trap a single light ray in a polygon.

Mitchell, Zachary, Gregory Simon, and Xueying Zhao. "Trapping light rays aperiodically with mirrors." Involve, a Journal of Mathematics 5.1 (2012): 9-14. (Journal link.)

Abstract. We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically.

To obtain a polygon, one would have to connect their disjoint segments into a path, but I think this would not be difficult.

But it is not possible to trap light rays from a continuum of directions:

Dawson, RJ MacG, B. E. McDonald, J. Mycielski, and L. Pachter. "Light Traps." (1996).

Yes, it is possible to trap a single light ray in a polygon.

Mitchell, Zachary, Gregory Simon, and Xueying Zhao. "Trapping light rays aperiodically with mirrors." Involve, a Journal of Mathematics 5.1 (2012): 9-14. (Journal link.)

Abstract. We construct a configuration of disjoint segment mirrors in the plane that traps a single light ray aperiodically, providing a negative solution to a conjecture of O’Rourke and Petrovici. We expand this to show that any finite number of rays from a source can be trapped aperiodically.

To obtain a polygon, one would have to connect their disjoint segments into a path, but I think this would not be difficult.

But it is not possible to trap light rays from a continuum of directions:

Dawson, RJ MacG, B. E. McDonald, J. Mycielski, and L. Pachter. "Light Traps." (1996). (PDF download.)