  Molecular lattice clocks enable the search for new physics, such as fifth
forces or temporal variations of fundamental constants, in a manner
complementary to atomic clocks. Blackbody radiation (BBR) is a major
contributor to the systematic error budget of conventional atomic clocks and is
notoriously difficult to characterize and control. Here, we combine infrared
Stark-shift spectroscopy in a molecular lattice clock and modern quantum
chemistry methods to characterize the polarizabilities of the Sr$_2$ molecule
from dc to infrared. Using this description, we determine the static and
dynamic blackbody radiation shifts for all possible vibrational clock
transitions to the $10^{-16}$ level. This constitutes an important step towards
mHz-level molecular spectroscopy in Sr$_2$, and provides a framework for
evaluating BBR shifts in other homonuclear molecules.
