No, there are surfaces that can't be expressed with implicit equations. For instance, a Möbius strip.
Off the bat this seems wrong as the set of rational surfaces is a subset of the set of algebraic surfaces. Furthermore, the most trivial embedding of the strip into poly-RP^3 (or rational-R^3) doesn't contain any base points. So finding the implicit form _should_ be a matter of technicalities. But I haven't really thought this through.
Some sort of mesh I would guess would be the input.
Hmmmm. Generally this is a topic in algebraic geometry, and it's far from being the simplest question in this relatively advanced branch of mathematics. If I understand you correctly, you're interested in an implicit representation of a piecewise linear mesh. If so then there's a slight problem, you'll have to use Abs in your implicit representation, which basically means that your surface is no longer an algebraic variety. Algebraic geometry deals with algebraic varieties, so you'll have to improvise if you want to use these methods to find your implicit equation.
However, if you're just sticking fingers at this problem, you can unify a few simple implicit surfaces by just multiplicating them in order to obtain a more complex implicit surface. This is basically how most people come up with these models, they just use the ready building blocks (cylinder, sphere, tori, cone, etc...) and mash them together.