I have an old laptop with an integrated Intel HD 520 graphics card which apparently doesn't support Vulkan.

On the Steam forums for this game, I found out that older versions of Gladius used OpenGL instead of Vulkan and it's possible to get that old version from Steam, where it works on old graphics cards like mine, which don't support Vulkan.

Is it possible to get such an old version of Gladius from GOG?

Nope, the oldest version is 4 months ago, on Galaxy or not.
I think even Slitherine had abandonned your old wanted build of years ago.

Yes, it was a bad move fot those who had bought the game but can't change their rig, as the requirement is not the same.
Those who could no longer play it should be proposed a refund.

Intel HD 520 has Vulkan support on Linux, assuming Mesa Vulkan drivers are installed. How to ensure they are installed depends on the Linux distribution in use (on Debian-based ones it is as simple as apt install mesa-vulkan-drivers).

On the other hand I have no clue about Windows drivers.

Intel states that the 520 does support Vulkan. Have you tried to simply update your driver first?