I wrote then, that I wasn’t 100% sure, I’ve just had a quick look and it seems that you need Vulkan-loader headers only to build Wine from source, but apparently don’t need it later when running:
$ ldd /opt/wine-devel/bin/wine
linux-vdso.so.1 (0x00007f082229c000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007f0822085000)
/lib64/ld-linux-x86-64.so.2 (0x00007f082229e000)
$ ldd /opt/wine-devel/lib/wine/x86_64-unix/winevulkan.so
linux-vdso.so.1 (0x00007f59e6eca000)
ntdll.so => not found
win32u.so => not found
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007f59e6bfa000)
/lib64/ld-linux-x86-64.so.2 (0x00007f59e6ecc000)
$ ldd /opt/wine-devel/lib/wine/x86_64-unix/win32u.so
linux-vdso.so.1 (0x00007f3176341000)
ntdll.so => not found
libm.so.6 => /lib/x86_64-linux-gnu/libm.so.6 (0x00007f3175f10000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007f3175d1b000)
/lib64/ld-linux-x86-64.so.2 (0x00007f3176343000)
$ ldd /opt/wine-devel/lib/wine/x86_64-unix/ntdll.so
linux-vdso.so.1 (0x00007f16bf0db000)
libunwind.so.8 => /lib/x86_64-linux-gnu/libunwind.so.8 (0x00007f16befdb000)
libc.so.6 => /lib/x86_64-linux-gnu/libc.so.6 (0x00007f16bede6000)
/lib64/ld-linux-x86-64.so.2 (0x00007f16bf0dd000)
liblzma.so.5 => /lib/x86_64-linux-gnu/liblzma.so.5 (0x00007f16bedb5000)
Further evidenced by the fact that none of the Wine packages depends on libvulkan1, but the build does depend on libvulkan-dev.