Any chain in a (sufficiently nice) wedge $X\wedge Y$ is homologous to a chain in $X$ plus a chain in $Y$. Now let $\alpha$ be a cocyle in $X$ and $\beta$ a cocyle in $Y$ and pull them back to cocycles $\tilde{\alpha},\tilde{\beta}$ on $X\wedge Y$.
By additivity and the first observation, it suffices to check that $\tilde{\alpha}\cup\tilde{\beta}$ vanishes on every simplex lying either in $X$ or in $Y$, but these get killed by $\tilde{\beta}$ or $\tilde{\alpha}$ resp. since they get pushed down to points.