Horizontal or Vertical?
Vertical foundations are just like wide monopoles or flag poles for the most part. Octagonal or circular foundation where anchorage of the vessel is the only thing that's more complicated than doing a standard building column.
Horizontal vessels can be different. Usually just a simple rectangular foundation. But, the loading can into things like "slide plates" that allow for temperature expansion / contraction on one side. That means the side with the slide plate gets designed to a lower lateral load (static friction coefficient of slide plate * dead weight on that pier). The other pier needs to be checked for that same force, but also for full seismic and wind.
I'm probably leaving something out because it's been 20+ years since I worked on one. But, I think I cover the basics.