Witness length independent $\exists$-Operator
You don't need a witness at all. You can allow the Turing machine to make a non-deterministic "guess" at any time. A string is accepted iff there is a sequence of guesses that make the Turing machine accept it, otherwise it is rejected.

Edit: If you want a definition which does not rely on the machine model, then it would be a bit awkward. Here is one example, which includes a cheat - using the associated function class. Given a (time) complexity $C$ and its associated function class $FC$, the allowed witness lengths are parametrized by $f \in FC$: the lengths are given by $n \mapsto |f(0^n)|$.