Is there a database/website with Butcher tableaus?

The three best sources of Butcher tableaus are, according to me, the reference books

• Numerical Methods for Ordinary Differential Equations, 3rd ed., J.C. Butcher,
• Solving Ordinary Differential Equations I, 2nd ed., Hairer, Norsett and Wanner and
• Solving Ordinary Differential Equations II, 2nd ed., Hairer and Wanner.

You will have to browse through the books but all three of them contain a huge number of tableaus with indications of the "method name" (like Dormand-Prince). Unfortunately, the Butcher book has a very limited index so you'll really need to browse the book.

There is also a comprehensive list on Wikipedia.

The Julia language extension DifferentialEquations.jl also has a large set of tableaus implemented. You can find the implementation of the tableaus here.

The Python package Nodepy has a number of tableaus, and will also generate tableaus for many different families of methods, including some that aren't traditionally viewed as Runge-Kutta methods but can be written that way (e.g., extrapolation methods and deferred correction methods).

Tableaus for many SSP methods can be found at http://sspsite.org; these are usually written in Shu-Osher form rather than Butcher form, since that's also how they are usually implemented.