Thanks to Nautarch's link, I have compiled the following chart:
\begin{array} {|r|r|r|r|r|r|r|r|r|r|r|}
\hline
&0th &1st &2nd &3rd &4th &5th &6th &7th &8th &9th &total \\
\hline
Acid &1 &2 &1 &3 &2 &0 &0 &1 &0 &2 &12\\
\hline
Cold &2 &4 &1 &3 &3 &1 &3 &1 &0 &2 &20\\
\hline
Fire &4 &5 &5 &5 &3 &2 &1 &3 &1 &2 &31\\
\hline
Force &1 &1 &1 &0 &0 &3 &1 &1 &0 &0 &8\\
\hline
Lightning &2 &3 &0 &5 &2 &0 &1 &1 &0 &2 &16\\
\hline
Necrotic &1 &3 &0 &2 &1 &1 &3 &2 &1 &0 &14\\
\hline
Poison &1 &2 &0 &0 &0 &1 &0 &1 &0 &1 &6\\
\hline
Psychic &1 &2 &1 &0 &2 &0 &0 &0 &1 &1 &8\\
\hline
Radiant &1 &2 &2 &3 &1 &2 &2 &0 &1 &0 &14\\
\hline
Thunder &2 &4 &1 &2 &1 &1 &0 &0 &0 &1 &12\\
\hline
Bludgeoning &1 &3 &3 &3 &4 &3 &2 &1 &2 &2 &24\\
\hline
Slashing &0 &1 &2 &0 &0 &0 &2 &0 &0 &0 &5\\
\hline
Piercing &0 &4 &2 &0 &1 &1 &1 &0 &0 &0 &9\\
\hline
\end{array}
A few notes:
I compiled it by searching the whole spell compendium by level and damage type, and just counting the results for each combination of level and damage type. I did my best to avoid counting the SRD and PHB version of spells multiple times (I.E. Only counting Bigby's Hand and Arcane Hand as one spell). The filters I used included all official sources up through Tales From the Yawning Portal as of the compilation of this chart.
Also, this chart only takes into account what was on the D&D Beyond compendium as of 7/12/2017 and thus won't include any new entries, unearthed arcana spells, or content that is missing from the DND:Beyond spell compendium. I also put my faith entirely in the categorization in D&D:Beyond. That is, if any spell actually does deal a form of damage, but D&D:Beyond didn't categorize that spell as dealing that damage type, then it would be omitted from this chart.
Finally, some spells either have multiple damage types, or have the option to choose from a variety of damage types. In both circumstances said spell was counted for every damage type that it could possibly inflict. The one exception to this is the wish spell which, due to its ability to copy any other spell of the 8th level or lower, has been omitted entirely.