A simple such characterization exists when one is working with proximity spaces instead of topological spaces. Suppose that $(X,\delta)$ is a proximity space with complete containment relation $\ll$. Then we say that a subset $Z\subseteq X$ is a proximally zero set if there is some proximity map $f:(X,\delta)\rightarrow[0,1]$ such that $Z=f^{-1}[\{0\}]$. On the other hand, the proximally zero sets are precisely the sets of the form $\bigcap_{n\in\omega}A_{n}$ for some sequence $(A_{n})_{n\in\omega}$ such that $\ldots A_{n+1}\ll A_{n}\ll A_{n-1}\ll\ldots\ll A_{0}$. Equivalently, a zero $A$ is a cozero set if and only if $A=\bigcup_{n\in\omega}A_{n}$ for some sequence $(A_{n})_{n\in\omega}$ with $A_{n}\ll A$ for all $n$. This characterization of proximally zero sets was first explicitly spelled out in my paper 2. Since in a completely regular space, a set is a cozero set if and only if it is a proximally cozero set in some compatible proximity, one can define a cozero set to be a set of the form $\bigcup_{n\in\omega}A_{n}$ such that there is a compatible proximity $\ll$ such that $A_{n}\ll A$ for all $n$.

Another similar internal characterization of zero sets is slightly more complicated and is given in 1. A collection $\mathcal{U}$ of open sets is a completely regular family if whenever $U\in\mathcal{U}$ there are sequences
$(U_{n})_{n\in\omega},(V_{n})_{n\in\omega}$ of elements in $\mathcal{U}$ so that
$U=\bigcup_{n\in\omega}U_{n}$ and $U_{n}\subseteq V_{n}^{c}\subseteq U$. In 1, it is shown that a set $U$ is a cozero set if and only if $U$ belongs to some completely regular family.

Also, I claim that one can also characterize the cozero sets internally in terms of a way-below relation. Suppose that $(X,\mathcal{T})$ is a topological space. Then define $U\prec V$ if and only if $\overline{U}\subseteq V$. Define $U\ll V$ if and only if there is some $(U_{r})_{r\in[0,1]\cap\mathbb{Q}}$ such that $U=U_{0},V=U_{1}$ and where $U_{r}\prec U_{s}$ whenever $r<s$. The following are equivalent:

$U$ is a cozero set.

There is a sequence $(U_{n})_{n\in\omega}$ of open sets with $U_{0}\ll U_{1}\ll U_{2}\ll\ldots U_{n}\ldots$ and where $U=\bigcup_{n\in\omega}U_{n}$

There exists some sequence $(U_{r})_{r\in[0,1)\cap\mathbb{Q}}$ of open sets with $U_{r}\prec U_{s}$ whenever $r<s$ and where $U=\bigcup_{r\in[0,1)\cap\mathbb{Q}}U_{r}$.

This characterization of cozero sets which does not refer to continuous functions is very useful in point-free topology since this characterization can easily be generalized to a point-free setting, and in point-free topology it is usually more convenient to work with open sets than it is to work with the pointfree analogue continuous real-valued functions. Hence, in point-free topology it is more convenient to define the cozero sets in terms of $\prec$ and $\ll$ than it is to define the cozero sets in terms of frame homomorphisms from the frame of real numbers $(\mathbb{R},\mathcal{T})$ to your original frame $L$.

If $X$ is a completely regular space, then the complete containment relation $\ll$ is completely below relation for the finest proximity compatible with the topology on $X$.

1Short proof of an internal characterization of complete regularity. Harald Brandenburg and Adam Mysior. Canad. Math. Bull. 27(1984), 461-462

2A generalization of the notion of a P-space to proximity spaces. J. Van Name
Volume 42, 2013. Pages 357–365