Do you, perhaps, have a file /etc/rc.modules, or a directory with that name?
If not, you might try and find a file /etc/rc.local, and just add modprobe yourmodule at the end.
/etc/rc.local file contains custom user commands that are executed on boot.
Previous solution works for most distros but not for you. You have some minimal rescue-like linux distro. This is why I suggest just appending modprobe yourmodule to the end of the /etc/inittab.
NOTE: THIS METHOD SHOULD BE AVOIDED IF YOU HAVE A NORMAL LINUX DISTRIBUTION LIKE UBUNTU, MINT, FEDORA, ARCH...
IF YOU USE A NORMAL DISTRIBUTION, PLEASE USE YOUR DISTRIBUTION'S WAY OF ADDING CUSTOM MODULES.