We study the existence of traveling wave solutions for a spatial susceptible-infected-recover (SIR) epidemic model with nonlocal dispersal and delayed transmission. The model incorporates convolution-type dispersal operators and a nonlocal time-delay incidence mechanism, which together lead to a non-cooperative and non-monotone traveling wave system. To overcome these difficulties, we construct an invariant cone on a large bounded interval and define a suitable integral operator associated with the traveling wave equations. Uniform a priori bounds and regularity estimates are established independently of the truncation parameter. By applying Schauder's fixed point theorem and a limiting argument, we obtain the existence of nontrivial traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium when the basic reproduction number exceeds one. Explicit upper and lower solutions are constructed to illustrate the applicability of the approach.