Suppose I have an Amiga computer without any working boot disks. Is it possible to trigger the built-in ROM monitor (RomWack or SAD, see https://theamigamuseum.com/amiga-kickstart-workbench-os/guru-meditation/) and then somehow send a program over the serial port to, say, write a disk image to disk? Would it require any hardware modifications to do this?
It is possible, but not simple. The ordinary boot process which brings a working machine up from cold to the prompt for a Workbench disk doesn't offer an opportunity to drop into the debugger.
A dodgy and potentially hardware-damaging approach is to cause an unexpected CPU exception by e.g. wiggling the trapdoor memory in an A500 -- execbase disappearing is sure to cause a crash -- or shorting the interrupt lines to cause an NMI. It'll crash, and while the power LED is blinking, you can take control over the serial port.
An arguably somewhat more "correct" approach is to add an expansion card of some sort which will hook the system and call Debug() on command. Building one from scratch would be more effort than it's worth, but existing cards could be presssed into service with similar effect.
Assuming an A500 or A1000, you can use an Action Replay cartridge. The one I had allowed me to format disks and install a custom bootblock, which gives enough of an edge to to bootstrap a working system from a machine that didn't come with Workbench disks. Earlier Action Replays might not be so featureful, but should at least let you break the running OS in such a way that it'll crash and drop into the debugger.
Another kind of expansion card is a hard disk controller. You can use an Amiga emulator to partition and install Workbench onto a hard disk which can then be transplanted into the real Amiga, which should then just boot off it. This is easier on a machine with built-in IDE since USB-IDE controllers are still easy to find, but SCSI is uncommon on modern machines. From here you can then just call Debug() directly, but since your bootstrap problem has already been solved, it's unnecessary.
You could also just get one of those GoTek USB-to-floppy adaptors, which also nicely sidesteps the problem of having no boot floppies.