Spent all evening figuring out those two jumpers on my relay board, sharing in case it helps

Mine did the same sort of thing on a chicken coop door project last winter. The relay would energize the moment I plugged the board in, before the Arduino even finished booting, then go quiet once my setup code ran. I was convinced the board was faulty and nearly ordered a replacement. Then I spotted the little plastic jumpers sitting across two sets of three pins and moved them, and suddenly everythign made sense. The input was being held low by the pin during boot, and because the board fired on low, the relay saw that as an on signal. Once I flipped the jumpers to high trigger and added a line in setup to force the pin high right away, the clicking stopped. One thing I'd add is to test the board on its own with a jumper wire between the input pin and ground or 5V before you even connect the microcontroller. That takes thirty seconds and tells you exactly which mode you're in, no code involved. Would have saved me a wasted evening of staring at my sketch wondering what I'd broken.

Worth adding that not every jumper on those boards does the same job. I bought a 4-channel one for controlling my garage workshop lights and assumed the little cap was a trigger selector like you describe. Moved it, nothing changed, relay behavior stayed exactly the same. Turns out on that board the jumper was bridging the relay coil supply to the pin that feeds the little isolation chip, and moving it actualy cut the coil power entirely, so the relays went dead. I put it back, left it alone, and just handled the active-low thing in code instead with a digitalWrite in setup before anything else runs. Took a few minutes with a multimeter in continuity mode and a magnifying glass over the board printing to figure out what the pads were actually connected to. So my take is slightly diferent from yours: before shuffling those jumpers around, trace what they connect, becuase some boards use that spot for power routing and not logic at all. Mine shipped with a sheet of paper that was just a bad photocopy of the pinout, no explanation of the jumper whatsoever, wich seems to be standard for these things.

Had a similar headache when I used one for my workshop automation. Even after fixing the trigger setup, I noticed the relay contacts would chatter a few times when it changed state. My code kept telling it to stay on, but it would quickly click off and back on for a split second. I realized I needed to add a small pause in the sketch after setting the pin state. It's not a deep theory thing, but just letting the switch settle before the next command runs defintely made the whole system stop acting crazy.

You gotta be careful though, even if the jumper setting is right, these budget modules are notorious for running on sketchy power. I've seen folks try to run a whole setup off a single 5V phone charger and then wonder why the trigger mode doesn't seem to stick. The coil needs clean, consistent power to snap reliably, expecially when the load is heavy. Back in the day, we always used proper linear regulators and big capacitors, not these little breadboard solutions that are just hoping the input voltage holds up. Try feeding the VCC pin directly from a robust supply and see if the behavior stabilizes.