$\newcommand\de\delta$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 $n\to\infty$.