Is the power set axiom essential for constructing L? Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
Axiom of Successor Cardinals: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$
where "$\leq$" refers to "cardinal smaller than or equal" relation, and $\kappa, \alpha$ range over von Neumann ordinals.

Can the resulting theory still interpret ZFC?

The idea is that if we can develop Gödel's constructible universe L inside this system, then this would interpret ZFC? So the power set axiom won't be essential for the development of L? 
 A: KP alone - which is vastly weaker than the theory in question - proves the sentence "For every ordinal $\alpha$, $L_\alpha$ exists," since it is strong enough to enable effective transfinite recursion. (We're passing to an unnecessarily weak subtheory, but it's worth noting.) The proof of this can be found e.g. in Barwise's book.
The condensation lemma, appropriately stated, can also be proved in KP; since our theory proves that successor cardinals exist, we get powerset in $L$. The proof that $L$ satisfies the rest of the ZFC axioms is the usual one.
So there is a uniform way to define in an arbitrary model $M$ of your theory an inner model (= transitive subclass containing all the ordinals of $M$) which is a model of ZFC + V=L.

Note the role of condensation in the above: condensation reduces powersets in $L$ to successor cardinals in $L$ (and hence a fortiori in any larger class). So it's not so much that we're avoiding powerset in building $L$, but rather that a very weak theory proves that powerset-in-$L$ is equivalent to successor-cardinals-in-$L$.
A: Let me not answer the question asked but add an important angle.
An L can be built already in ATR_0, which is the weakest theory that can do it convincingly (some coding involved).
You can't guarantee Powerset in that L, but it can well happen that this L acquires lots of uncountable cardinals. (All ordinals were "countable" in the initial model of arithmetic, but after the extraction of L, many original bijections between ordinals and N were left outside.) 
I guess the best source for this is Simpson's "Subsystems of Second Order Arithmetic", parts VII.3 and VII.4.
Perhaps what is also very relevant to your thoughts is something called "the Feferman-Leví model" discussed on page 295 of Simpson's book and elsewhere.
