How could I approach determining if this 2D system represented as a 2D summation formula is linear?
I'm actually starting this out not knowing the answer, although I have my suspicions.  I think it'll be a linear system, but I'm not sure.  Here's your starting point: you define the system $y = h(x)$ as:
$$y = h(x) \mid y(m,n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} x(k_1,k_2)$$
So let
$$\begin{aligned}
y_1(m,n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} A\,x_1(k_1,k_2) = h\left(A\, x_1 \right) = A h(x_1) \\
y_2(m,n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} B\,x_2(k_1,k_2) = h\left(A\, x_2 \right) = B h(x_2)
\end{aligned}$$
Then
$$
y_1(m,n) + y_2(m, n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} A\,x_1(k_1,k_2) + \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} B\,x_2(k_1,k_2)
$$
Because the summations have identical limits, we can bring everything under one pair of summations:
$$
y_1(m,n) + y_2(m, n) = \sum_{k_1=-\infty}^{m} \sum_{k_2=-\infty}^{n} \left ( A\,x_1(k_1,k_2) + B\,x_2(k_1,k_2) \right)
$$
but this is just $y_1 + y_2 = h(A\,x_1 + B\,x_2)$.  You already know that $y_1 = A h(x_1)$ and $y_2 = B h(x_2)$, so linearity is proved.
(Which is what I thought -- I was going to just dismiss your original problem as &quot;summations are linear, go away!&quot; -- then I realized I wasn't sure).