Do tornadoes have eyes? In the movie Into the Storm (2014) near the end, storm chaser Pete sees the eye of a massive tornado.

In 1928 (real life), Will Keller was in his barn when a huge tornado passed through. He reported seeing an opening in the center of the tornado "about 55 feet (17 meters) across and extended some 2,500 feet (762 meters) up." Source

Is there any "official" documentation or evidence that tornadoes, especially stronger ones, have eyes like a hurricane does? Or is it just an urban legend?
Yes, if one takes the common meaning of the term "eye of the storm" to be the area of relatively low wind speed near the center of the vortex, most tornadoes can be said to have eyes. Cyclostrophic balance describes a steady-state, inviscid flow with neglected Coriolis force:

$$
\dfrac{v^2}{r} = -\dfrac{1}{\rho}\dfrac{\partial p}{\partial n}
$$  

where centripetal force balances radial pressure gradient. Here, $v$ is tangential wind speed, $r$ distance from vortex center, $\rho$ is air density, $p$ is atmospheric pressure and $n$ is the radial direction pointing inward. From here, tangential wind speed is simply:

$$
v = \sqrt{-\dfrac{r}{\rho}\dfrac{\partial p}{\partial n}}
$$

suggesting that $v\to0$ when $r\to0$. While the flow in tornadoes is highly non-stationary and subject to friction, this idealized model shows why there must exist an "eye" inside a vortex or an area of closed circulation. This "eye" may or may not be easily recognized by a hypothetical human observer inside a tornado.