Extremely accurate state-of-the-art potentials for more than 20 different diatomic molecules (including excited states for many of them!) are given here.
Bob Le Roy has a reputation for building some of the best small molecule potentials in the world, and in recent years he has started to make them available online. But these are not just a Taylor series around equilibrium, they are analytic functions covering the entire landscape of possible geometries, and for diatomics the specific function is given as $V(r)$ here.
You can plug the parameters from the given publications into the $V$ function, and then Taylor expand the global analytic function into a Taylor series if you want, but the original (non-Taylor expanded) function will be more accurate globally.
If the molecule doesn't yet have a potential available, you can build one ab initio
(which is just the Latin phrase for "from scratch") using a program such as the free version of MOLPRO
which can be found in the row of the linked webpage saying 0.00 EUR.
The input for making potentials in MOLPRO is here.
If you have specific molecules you are interested in, and you can tell us what you wish to do with these potentials, I can try to direct you to the best potentials for the ground and excited states for the particular molecules you ask for. There is no comprehensive database that has all potentials in one place.