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 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.