Yes, by definition $$ \iint f(x,y)dxdy = \lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum f(\xi_i,\psi_j)\Delta x_i \Delta y_j$$
Now $$\iint cf(x,y)dxdy = \lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum cf(\xi_i,\psi_j)\Delta x_i \Delta y_j $$
Now, use the properties of Sums and limits to obtain:
$$\lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum cf(\xi_i,\psi_j)\Delta x_i \Delta y_j = c\left(\lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum f(\xi_i,\psi_j)\Delta x_i \Delta y_j\right) $$
Therefore,
$$\iint cf(x,y)dxdy = c\iint f(x,y)dxdy$$