I think this isn't hard if you don't care at all about covariance structure or regularity of $Z_t$.  For any given $t$, your formula defines a valid cumulative distribution function, so such a random variable $Z_t$ exists.  Now [this answer to another question][1] says you can construct an uncountable family of independent random variables, so this is enough.  I don't know how that construction works, so an alternative is to construct independent random variables $Z_t$ for rational $t$, and then define $Z_t$ for irrational $t$ as an inf or sup.

If you want $Z_t$ to have, for example, continuous sample paths, then it's a harder question.


  [1]: https://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20834#20834