= Solution
{c}
Normalize the <generating functional> by its value with all sources zero, and take the connected part of the <time-ordered correlation function>. Source differentiation of the interaction exponential is equivalent to the <Dyson series> followed by <Wick's theorem>. Write $D(x-y)=\langle T\phi(x)\phi(y)\rangle_0$ and $F_{ij}(x-y)=\langle T\psi_i(x)\bar\psi_j(y)\rangle_0$ temporarily, so that every contraction has an unambiguous meaning.
The second-order <Dyson series> contributes
$$
\frac{(ig)^2}{2!}\int d^4x\,d^4y\,
\langle T\phi(x_a)\phi(y_a)\phi(x)\phi(y)
[\bar\psi_i(x)\psi_i(x)][\bar\psi_j(y)\psi_j(y)]\rangle_{0,c}.
$$
There are two connected external <scalar field> attachments:
$$
D(x_a-x)D(y_a-y)+D(x_a-y)D(y_a-x).
$$
They become equal after exchanging the integration variables $x,y$, cancelling $2!$. The connected <fermion> contractions have the <closed fermion loop sign>:
$$
\langle T[\bar\psi_i(x)\psi_i(x)][\bar\psi_j(y)\psi_j(y)]\rangle_{0,c}
=-F_{ij}(x-y)F_{ji}(y-x)
=-\operatorname{tr}[F(x-y)F(y-x)].
$$
The minus sign can also be checked directly: the pairing of the first with the fourth field and the second with the third has positive pairing parity, but $\langle T\bar\psi_i(x)\psi_j(y)\rangle=-F_{ji}(y-x)$. Multiplying it by $(ig)^2=-g^2$ leaves a positive result:
$$
G_c^{(2)}(x_a,y_a)=g^2\int d^4x\,d^4y\,
D(x_a-x)D(y_a-y)\operatorname{tr}[F(x-y)F(y-x)].
$$
A contraction of the two external fields with each other times a <vacuum bubble> cancels against the normalization. Separate <fermion> <tadpole integrals> at $x$ and $y$ belong to a product of one-point functions, removed by taking the connected part.
To match the source's <generating functional> convention, distinguish its displayed kernels from contractions. The positive $iJ\Delta J/2$ exponent gives $D=-i\Delta$. With the source insertion order in equation (5), the positive $i\bar\eta S_F\eta$ exponent similarly gives $F=-iS_F$. The two <scalar propagators> and two <Dirac propagators> supply a total phase $(-i)^4=1$. Thus, writing $S=S_F$ as in the requested formula,
$$
\boxed{G_c^{(2)}(x_a,y_a)=g^2\int d^4x\,d^4y\,
\Delta(x_a-x)\Delta(y_a-y)\operatorname{tr}[S(x-y)S(y-x)].}
$$
The <matrix trace> closes the <spinor> indices; no extra factor of two or additional <symmetry factor> remains for a <Dirac field> with this <Yukawa interaction>.
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