This is not an answer, but too long for a comment, and hopefully of some use.
I believe the theorem you mention should be true for any 5-dimensional Poincaré duality space $X$. You can define the linking form on the torsion group $H^3(X,\mathbb{Z})_{\rm tor}$ (which is isomorphic $H_2(X, \mathbb{Z})_{\rm tor}$ via the Poincaré duality structure) by setting $(x,y) = x' \cup y$, where $x' \in H^2(X,\mathbb{Q}/\mathbb{Z})$ maps to $x$ under the boundary map $\partial: H^2(X,\mathbb{Q}/\mathbb{Z}) \to H^3(X,\mathbb{Z})$ associated to the short exact sequence $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$.
It should then be true that $(x,x) = (x,\partial \overline{w}_2) = x'\cup \partial \overline{w}_2$, where $\overline{w}_2$ is the image in $H^2(X,\mathbb{Q}/\mathbb{Z})$ of the second Stiefel-Whitney class $w_2 \in H^2(X,\mathbb{Z}/2)$ of the Spivak normal bundle. This would imply that the linking form is alternating if and only if $\partial \overline{w}_2$ vanishes in $H^3(X,\mathbb{Z})_{\rm tor} \subseteq H^3(X,\mathbb{Z})$, which, in turn, is equivalent to $w_2$ lifting to $H^2(X,\mathbb{Z})$. In the case of smooth manifolds this is the same as admitting a ${\rm spin}^{\mathbb{C}}$-structure.
For simply-connected 5-dimensional Poincaré duality spaces this is proved in Stöcker's paper from 1982 entitled "On the structure of 5-dimensional Poincaré duality spaces", but I was not able to find a proof for the general case.
It's possible that there is a small gap in the literature here, so it sounds to me that if you have a proof (in either the smooth, topological or Poincaré setting) it would be great of you write it up and publish it. I would certainly be interested to read it.