Ok, I disabled the fsck root filesystem check in /etc/rc.d/rc.S and I was able to boot on the computer, which handles the USB memory stick as /dev/sdb1. Everything seemed to be allright. It's ugly and baaad, but it's ok as a temporary solution.
But the biggest problem still lies ahead - LILO is configured to use /dev/sda1, if I boot on a computer, which handles the USB memory stick as /dev/sdb1, I need to enter root=/dev/sdb1 kernel parameter. Any ideas how to "autodetect" the device name of device, from which the LILO is running?
I have to admit that I didn't read the Vector Linux Live init scripts yet, so maybe it needs a little work. Or maybe it's a matter of /boot directory and some ramdisk - in that case i'm doomed.