Step 1: Understanding the reaction mechanism.
The reaction starts with ethene (C$_2$H$_4$), which reacts with an electrophile to form bromoethane (C$_2$H$_5$Br) in the first step. The appropriate reagent to add a bromine atom to ethene is HBr. In the second step, the bromoethane reacts with a nucleophile to form ethyl propionate. The nucleophile in this case is C$_2$H$_5$COOAg (silver acetate), which displaces the bromine atom to form the ester.
Step 2: Analyzing the options.
(A) Br$_2$/AlBr$_3$, C$_2$H$_5$ONa: This is not correct, as the reaction does not involve a bromine addition via AlBr$_3$ in the first step.
(B) Br$_2$/AlBr$_3$, C$_2$H$_5$COOAg: This reagent combination is also incorrect, as the addition of Br$_2$ does not occur in this mechanism.
(C) HBr, C$_2$H$_5$ONa: This is not correct because sodium ethoxide (C$_2$H$_5$ONa) would not be the nucleophile in the second step for ester formation.
(D) HBr, C$_2$H$_5$COOAg: This is the correct combination, as HBr adds across the double bond of ethene to form bromoethane, and then silver acetate (C$_2$H$_5$COOAg) displaces the bromine atom to form ethyl propionate.
Step 3: Conclusion.
The correct answer is (D) HBr, C$_2$H$_5$COOAg, as it correctly describes the reagents for the two steps in the reaction mechanism.