= Solution
Use the notation $\Delta_{12}=\Delta(y_1-y_2)$, $\Delta_{xz}=\Delta(x-z)$ and $\int_{xz}=\int d^Dx\,d^Dz$. The order-$\lambda$ term vanishes because it has five <fields>. At order $\lambda^2$, enumerate the eight-field <Wick contractions>, whose expansion denominator is $2!(3!)^2=72$.
If the two external <fields> contract with each other, the remaining <vacuum diagrams> are a pair of cubic <vertices> joined by three lines, with six <Wick contractions>, and a pair joined by one line with a <tadpole diagram> at each end, with nine <Wick contractions>. Define their combined integrated weight by
$$
V_2=\int_{xz}\left[\frac1{12}\Delta_{xz}^3+\frac18t^2\Delta_{xz}\right].
$$
Thus $Z/Z_0=1+(-i\lambda)^2V_2+O(\lambda^4)$. The numerator is
$$
\begin{aligned}
\frac{\mathcal N_2}{Z_0}
={}&\Delta_{12}\left[1+(-i\lambda)^2V_2\right]\quad+\\
&(-i\lambda)^2\int_{xz}\left[
\frac12K_1(x)K_2(z)\Delta_{xz}^2
+\frac12K_1(x)K_2(x)\Delta_{xz}t
+\frac14K_1(x)K_2(z)t^2\right]+O(\lambda^4).
\end{aligned}
$$
The three latter terms have respectively 36, 36 and 18 <Wick contractions>. Together with the six and nine <Wick contractions>, they exhaust all $7!!=105$ pairings.
Dividing by $Z/Z_0$ cancels the two <vacuum diagrams>, giving
$$
\boxed{\begin{aligned}
\langle T\phi(y_1)\phi(y_2)\rangle={}&\Delta_{12}+(-i\lambda)^2\int_{xz}\Bigl[
\tfrac12K_1(x)K_2(z)\Delta_{xz}^2\\
&+\tfrac12K_1(x)K_2(x)\Delta_{xz}t
+\tfrac14K_1(x)K_2(z)t^2\Bigr]+O(\lambda^4).
\end{aligned}}
$$
The first correction is the two-line <bubble diagram>, with <symmetry factor> $S=2$. The second is the connected <tadpole contribution to a cubic-theory two-point function>: both external lines meet one <vertex>, joined by a single line to a second <vertex> bearing a <tadpole diagram>; its <symmetry factor> is also $S=2$. Interchanging the two integrated <vertices> is already included in the 36 contractions and gives no extra factor.
The final term is the <disconnected Feynman diagram> consisting of two separate one-point <tadpole diagrams>, with coefficient $1/4$. It survives <cancellation of vacuum bubbles>, since each component touches an external point. Indeed it is precisely the leading $\langle\phi(y_1)\rangle\langle\phi(y_2)\rangle$. It would disappear from the connected <two-point correlation function>, but the question defines the full <correlation function>. No <normal ordering> or <tadpole subtraction> of the interaction was specified. All terms are depicted in <3/b/solution>, and all <spacetime> integrals remain unevaluated.
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