Note: you might be having the same problem I helped solve in this thread - one issue was that the compiler needed to be run as two commands, first compile and then link.
You shouldn’t need to set any flags to get Vulkan support (it’s build in).
See the GLFW Vulkan guide for what you do once you have a working set of GLFW libraries. Note that you need the LunarG SDK, and I don’t know how easy that is to use with MinGW.
Since Visual Studio Community edition is free to use for home and small studio use, and you can run compiles from the command line, you could try that if you get into problems.