according to TechPowerUp website for the specs on the HD 5850 it doesn't support Vulkan API: ATI Radeon HD 5850 Specs | TechPowerUp GPU Database
The reason why is because AMD doesn't support that GPU with Driver updates anymore. Thus you only have a choice of two last AMD Drivers for that GPU card (2015 & 2016) unless the manufacturer of your GPU card has newer drivers.
https://www.amd.com/en/support/graphics/amd-radeon-hd/ati-radeon-hd-5000-series/ati-radeon-hd-5850
EDIT: Vulkan is enabled by the GPU driver. Vulkan wasn't part of the AMD Driver yet in those two drivers.
From Vulkan:
AMD Radeon Software Crimson Edition 16.3 and later support the following AMD APUs and Radeon™ GPUs1 based on the Graphics Core Next architecture on Windows® 7, Window® 8.1, Windows® 10, and Linux®