$\newcommand\de\delta$Let $a:=Ef(X_1,Y_1)$. Replacing $f$ by $f-a$, assume without loss of generality (wlog) that $Ef(X_i,Y_j)=0$ for all $i,j$. Let $$Z:=\frac1{n^2}\sum_{i,j=1}^n Z_{ij},\quad Z_{ij}:=f(X_i,Y_j).$$ Then, letting $\cdot$ denote the inner product, we get
$$\begin{aligned} n^4E\|Z\|^2&=nE\|Z_{11}\|^2 \\ &+2n(n-1)(EZ_{11}\cdot Z_{12}+EZ_{11}\cdot Z_{21}) \\ &+n(n-1)(EZ_{12}\cdot Z_{12}+EZ_{12}\cdot Z_{21}) \\ &+n(n-1)(n-2)(EZ_{12}\cdot Z_{13}+EZ_{21}\cdot Z_{31} +EZ_{12}\cdot Z_{31}+EZ_{21}\cdot Z_{13}) \\ &\le Cn^3 \end{aligned}$$ for a certain real $C>0$ depending only on $f$ and the distributions of $X_i,Y_j$.

So, for any real $\de>0$, $$P(\|Z\|\ge\de)\le\frac{E\|Z\|^2}{\de^2}\le\frac C{n\de^2}\to0$$ as $n\to\infty$.

$\newcommand\de\delta$Let $a:=Ef(X_1,Y_1)$. Replacing $f$ by $f-a$, assume without loss of generality (wlog) that $Ef(X_i,Y_j)=0$ for all $i,j$. Let $$Z:=\frac1{n^2}\sum_{i,j=1}^n Z_{ij},\quad Z_{ij}:=f(X_i,Y_j).$$ Then, letting $\cdot$ denote the inner product, we get
$$\begin{aligned} n^4E\|Z\|^2&=nE\|Z_{11}\|^2 \\ &+2n(n-1)(EZ_{11}\cdot Z_{12}+EZ_{11}\cdot Z_{21}) \\ &+n(n-1)(EZ_{12}\cdot Z_{12}+EZ_{12}\cdot Z_{21}) \\ &+n(n-1)(n-2)(EZ_{12}\cdot Z_{13}+EZ_{21}\cdot Z_{31} +EZ_{12}\cdot Z_{31}+EZ_{21}\cdot Z_{13}) \\ &\le Cn^3 \end{aligned}$$ for a certain real $C>0$ depending only on $f$ and the distributions of $X_i,Y_j$.

So, for any real $\de>0$, $$P(\|Z\|\ge\de)\le\frac{E\|Z\|^2}{\de^2}\le\frac C{n\de^2}\to0$$ as $n\to\infty$.

In principle, one can bound other (higher) moments of $\|Z\|$ to get better bounds on $P(\|Z\|\ge\de)$, but that is much harder technically -- cf. the 41-page treatment by Adamczak and Latała of the case of U-statistics with completely degenerate kernels. Note that $Z$ is a V-statistic (or order $2$), which is harder to deal with than with U-statistics of the same order. However, Adamczak and Latała deal with arbitrary orders.