The Materials Project and other materials databases have a nice periodic table on their home page that allows you to search for compounds. For some materials, they give an empirical $U$ value used to model the bandstructure, and a reference to the paper. However, this information is available for a limited amount of materials.
Is there a databse or repository where one can find $U$ values obtained by experiment or determined ab-initio for metals across the periodic table?
Unfortunately, I don't believe such a resource exists. One should keep in mind that it's not just a matter of a single U value for every metal, of course. The "best" empirical U value will depend on the property of interest. Perhaps more problematic, not every metal environment is made equal. Different oxidation states, coordination environments, and overall electronic structures will inevitably influence the empirically ideal U value as well as any ab initio U value, making it hard to have a one-value-fits-all approach for a given metal.
This variety and application-specific nature is probably the major reason why such a database doesn't exist right now, to the best of my knowledge. That being said, I think it would be an amazing resource if such a database did exist, even if there were a variety of U values categorized by material and property.
Beyond the band structures, it's worth mentioning that a commonly used set of U values are those taken from this work by Ceder and coworkers to reproduce the oxidation energies of transition metal oxides. This is what is used throughout the Open Quantum Materials Database to try and achieve more accurate formation energies.
