Yes, we need the axiom of countable choice for choosing such bijections.
If we can choose a bijection for each finite set, then their union is countable, since we can map it into $\omega\times\omega$ in the obvious way.
An example of a countable set of finite sets, even pairs, whose union is not countable, if so, is an example of a countable family of finite sets that we cannot choose uniformly bijections for all pairs.
For example, consider the mathematical realization of Russell's socks example. It is consistent that there is a countable family of pairs, that does not admit a choice function. If the union of these pairs would have been countable, or even linearly orderable, then we could have chosen the least one from each set. Since we cannot do that (within that particular model of set theory), that union is not countable, and we cannot choose an enumeration for each pair uniformly.
It should, perhaps, be pointed out that in the case of finite sets this is weaker than the axiom of countable choice. This is not a trivial theorem, but an important nonetheless.
Countable choice implies the countable union of countable sets are countable implies every countable family of finite sets admits a choice function (is equivalent to your question about choosing enumerations).
None of the implications are reversible.