Rates of Keto-Enol Tautomerism
This is not a database, and I don't know about how small molecules you are looking for, I found that article on Jacs Kinetics and thermodynamics of keto-enol tautomerism of simple carbonyl compounds: an approach based on a kinetic study of halogenation at low halogen concentrations

If you can access the file it will be better for you, if not I let you some matrix of values. I hope it can help ! I give you them in the same the order of the publication.

I added some other publication at the end.















Links 

Tautomerization of 2-Acetylcyclohexanone. 1. Characterization of Keto−Enol/Enolate Equilibria and Reaction Rates in Water

Kinetic studies of keto-enol and other tautomeric
equilibria by flash photolysis (a lot of constants in this one)