As an addendum to Joel Hamkins answer: the weaker assertion (`Transitive Containment') that every set x is contained in a transitive set (not necessarily its transitive closure) is not provable in ZF - Replacement (sometimes known as Z Zermelo-set theory).
As Joel says in his answer we need to collect together the results of taking successive $\bigcup$. For this $\Sigma_1$-Replacement is more than enough (if AxFoundation is formulated in the right way for $\Pi_1$ classes).
As a second addendum (as the discussion continues) one should remark that we have in full ZF the $\in$-recursion theorem: thus we may define the class function $x\rightarrow TC(x)$ within ZFC (no need for NBG) via the recursion scheme $$TC(x)= \bigcup [TC(y) : y\in x] \cup x $$. The function defined has just the same status as $\alpha \rightarrow V_\alpha$ (which is also defined by such a recursion scheme). The former guarantees the existence of $TC(x)$ for any $x$, the latter of $V_\alpha$ for any $\alpha$. Neither function is problematic for the full ZF-set theorist.
The original question seems to be spurred on by the fact that we do not have an obvious {\em explicit definition} for $TC(x)$ in the way that we do for say ordered pairs. However that is often the case: turning recursive definitions into explicit definitions may not be possible.