This is a straightforward generalization of the customary definition of totally bounded sets on uniform spaces:

Definition Reloid $f$ is totally bounded iff for every $E \in \mathrm{GR}\, f$ there exists a finite cover $S$ of $\mathrm{Ob}\, f$ such that $\forall A \in S : A \times A \subseteq E$.

See here for definitions and notation.

I don’t know which interesting properties totally bounded spaces have (except of their connection to compact spaces, but at the time of writing this compactness of funcoids is not yet properly defined).