The final sentence of Theorem 1 of Browder's Remark on the Poincaré Duality Theorem states that a five-dimensional space $X$ with Poincaré duality has $H_2(X) = F + T + T + \mathbb{Z}_2$, where $F$ is free and $T$ is torsion, if and only if $w_3 \neq 0$. If $b$ denotes the linking form and $x$ is the generator of $\mathbb{Z}_2$, then $b(x, x) = \frac{1}{2}$ so $b$ is not alternating. Therefore, if $b$ is alternating, then $w_3 = 0$ which is equivalent to $w_2$ having an integral lift (which is equivalent to spin${}^c$ in the smooth case).

I don't know of a reference for the converse. However, if $H_2(X)$ has no two-torsion, then it is spin${}^c$ and the fact that the linking form is alternating follows from the classification of non-singular anti-symmetric linking forms; see here.

The final sentence of Theorem 1 of Browder's Remark on the Poincaré Duality Theorem states that a five-dimensional space $X$ with Poincaré duality has $H_2(X) = F + T + T + \mathbb{Z}_2$, where $F$ is free and $T$ is torsion, if and only if $w_3 \neq 0$. If $b$ denotes the linking form and $x$ is the generator of $\mathbb{Z}_2$, then $b(x, x) = \frac{1}{2}$ so $b$ is not alternating. Therefore, if $b$ is alternating, then $w_3 = 0$ which is equivalent to $w_2$ having an integral lift (which is equivalent to spin${}^c$ in the smooth case).

I don't know of a reference for the converse. However, if $H_2(X)$ has no two-torsion, then it is spin${}^c$ and the fact that the linking form is alternating follows from the classification of non-singular anti-symmetric linking forms; see here.