How are annihilation/creation operators used to reach an external state of $|0 \rangle$ in an $S$-matrix? I'm trying to understand how to compute the $S$-matrix element for $\phi \phi \to \phi \phi$. In
"Peskin and Schroeder's  Ch. 4.6". I'm lead to believe that, in $\phi^4$ theory,
$$ S = \langle p_3 p_4 | N (-\frac{i\lambda}{4!})\int d^4 x \phi^4 (x)| p_1, p_2\rangle, \tag{4.92}$$
It is then stated:
"Since the external states are $|0\rangle$  (...) we can use an annihilation operator from $\phi(x)$ to annihilate an initial-state particle, or a creation operator from $\phi(x)$ to produce a final-state particle. For example:"
$$\phi(x)|p\rangle = e^{-ip\cdot x}| 0 \rangle , \hspace{5mm}\langle p| \phi(x) = \langle 0| e^{ip\cdot x}\tag{4.94}$$

Intuitively, to simplify the $(4.92)$ and reach an external state of $|0 \rangle$, I've substituted each of the 4 $\phi(x)$ according to $(4.94) $, but I'm not certain of this as  I have two initial and two final states instead of the single state presented in $(4.94) $.
Another reason why I believe my answer is incorrect is due to the need for commutations in the $\phi^4$ case, as shown in (4.95) on the link above.
My answer would therefore be:
$$ S = \langle 0| N (-\frac{i\lambda}{4!})\int d^4 x\ e^{i(p_3 +p_4)\cdot x}  e^{-i(p_1 +p_2)\cdot x}| 0\rangle$$
Is my take on this correct?
 A: To be 100% correct, you should really use the LSZ formula to connect asymptotic in/out states to $n$ point vacuum Greens functions of time ordered fully interacting operators.  You should then apply the Gell-Mann--Low theorem to connect the $n$ point vacuum Greens function of the fully interacting fields to a ratio of time ordered vacuum Greens functions of the interaction picture fields.  Finally, one expands the result in a Dyson series and evaluates using Wick contractions.
As a simple hack that'll work to leading order, note that at leading order $|\vec p\rangle_{in} = |\vec p\rangle = a^\dagger(\vec p)|0\rangle$ and similar for ${}_{out}\langle \vec q|$.  Then simply apply the commutation relations for raising and lowering operators; depending on your conventions, something like $[a(\vec p),a^\dagger(\vec q)] = (2\pi)^m2E_{\vec p}\delta^{m}(\vec p - \vec q)$ for a field theory in $m+1$ spacetime dimensions.  Remember to use the facts that $a(\vec p)|0\rangle=0=\langle0|a^\dagger(\vec q)$.
