I would have expected molecules like IF5 to have 4, with two planes along the bond axes, and two planes bisecting the bonds. This is similar to how square planar molecules have four vertical mirror planes. However, apparently there are only two planes. Why is this? What am I misunderstanding?
Either your book is wrong or you are misunderstanding what it is trying to say.
There are 4 vertical planes of symmetry.
See for example GROUP THEORY AND ITS APPLICATIONS IN CHEMISTRY, SECOND EDITION , section 1.5.2
Vertical planes are further classified into two, namely $\sigma _v$ and $\sigma _d$ The $\sigma _v$ planes fall along the bonds and the $\sigma _d$ planes bisect the bond angles in molecules. The subscripts v and d stand for vertical and diagonal. However, both $\sigma _v$ and $\sigma _d$ are vertical planes