I'm not the best at math, but I'm wondering if there is a formula to figure out the thickness of a lens from only knowing the focal length and diameter.
You need to know more than those values, but they're necessary. The other thing you need to know is the refractive index of the material the lens is made from.
Knowing these three values, ray tracing computations can give you the exact profile of both surfaces of a single element, which will necessarily tell the minimum thickness. However, within reason, you can add an arbitrary amount of extra thickness to the lens without affecting its focal length (though you will change the curvature of at least one of the surfaces because the extra thickness is extra distance separating the first surface from the second -- and the two surfaces are independent refracting elements).
The computation isn't really a simple formula, it's a set of rules for how light refracts at an interface, and except for very simple cases, it requires software to get from a set of parameters to a final lens prescription. Ray tracing software of a general sort isn't available for free, but I recall reading about a limited version that was freely available (limited to something like four or five elements and only a few glasses -- it also handles dispersion, so you could calculate chromatic aberration corrections).