Is there a method or plastic/polymer to form an injection molded
plastic part that is smooth, distortion free and have good reflective
qualities? Reflective qualities such as the glancing reflection of a
smooth body of water or pane of glass. Typically, plastic shrinks and
distorts when cooled.

Not my speciality but perhaps observations may be of assistance.

Injection moulding *can* be very precise for small objects, where this
is the prime requirement, by maintaining the injection pressure whilst
the mould is cooled to hand temperature. Filled polymers also exhibit
far lower distortion and contraction but lack optical qualities of
course.

LED encapsulations are pretty precise and fine finished straight from
the mould but they use resin, presumably low shrinkage. Dunno what is
done for release agent.

But do you have to use plastic? There are low melt temperature metals
which don?t shrink.

Best

Polycarbonate is smooth, distortion free (as used in optics in cameras
and spectacle lenses). Uneven shrinkage in moulding is usually down to
bad tooling design where the flow of plastic isn't good (flow lines)
or the moulding has too many ribs of varying sizes and distribution
(shrinkage marks). Reflective quality is good, as polycarbonate can
exhibit a high gloss finish. Size is influential, larger objects will
distort more than small objects. I believe polycarbonate can be
precision micro engineered for parts such as surgical tools. This
contribution is from memory so facts needs to be verified.

Of course there is. The most common example is optical discs (CD, DVD)
that are formed by injection molding acrylic (previously) and
polycarbonate (presently). The smoothness or uniformity of surface has
to be such that pits of depth ~100 nanometers are easily detected. The
surface flatness is comparable to the best mirrors. The discs are
cooled in the mould to avoid problems associated with flow of uncooled
liquid ("smearing") and you can see that the process is highly
successful from the fact that CDs and DVDs are produced in the
billions.