We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $L_{\phi }$, namely when the generating $N$-functions fulfill ${\mathrm{\Delta }}^{\prime }$ and ${\mathrm{\Delta }}_3$-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.