The closest to your old AMD APU Processor A10-5800K is AMD's A10-5800B with HD 7660D which should be compatible with your processor Integrated Graphics.
But unfortunately your processor is too old to support any AMD driver newer than 2016 BETA. Drivers newer than 2016 BETA starts supporting Vulkan.
So, unless you find some modified newer AMD Drivers that will be supported on your laptop the only way to get Vulkan working is by upgrading either your APU processor or laptop to one that is still supported by AMD.
