You may have it backward. You can run a kernel and use busybox as the init process.
The way that works is that the kernel+initrd are loaded, the kernel then creates a memory resident filesystem restored from the initrd. The kernel then execs a program for init - and that CAN be any program at all. When that program exits the kernel will halt (or panic, depending how it exits).
Busybox makes a good init - it has many of the core commands builtin, so there is no need for external storage for those.
reference:
https://busybox.net/about.html
The commands are listed at:
https://busybox.net/downloads/BusyBox.html