I'm just curious about this. Technically, can a vector function be considered a composite function? Are they equivalent? For example, determining the domain of a vector function, will it be the same as a composite function of its components? For example
$ \vec{r}(t) = \langle f(t), g(t), h(t) \rangle \equiv (f\circ g \circ h)(t) $
(Comment turned answer)
Recall that given a function $f:A\rightarrow B$ and $g:B\rightarrow C$, the composite $g\circ f$ is a function from $A$ to $C$ defined by $(g\circ f)(a)=g(f(a))$.
Applying this to your problem, each of $f,g,h$ are real-valued functions of a real variable. Then applying the above twice, we see that $f\circ g \circ h$ is a real-valued function of a real variable as well.
However, $t\mapsto \langle f(t),g(t),h(t)\rangle$ takes values in $\mathbb{R}^3$ rather than $\mathbb{R}$, so they won't be equal (or even have the same codomain/range).