I don’t know of any way to totally wipe out chipmunks. Even a serious poisoning campaign normally doesn’t totally get them all. The best you can do is to try to get the numbers under control.
Cats take some time but they can be quite effective as long as they have access to the area where the chipmunks are. Some dogs, say a small terrier, can be quite effective too. But if you keep the animal confined where it can’t get at them they don’t do much good. I could see locking a cat in that greenhouse though I don’t know how much damage the cat might do.
Snakes will eat them, but they are less effective. A snake will eat one then stay pretty quiet while it digests it. They don’t kill just to kill, like some things will. I like having snakes around to help keep the number of rodents down, but snakes don’t get them all. It needs to be a reasonably big snake for chipmunks. Some people just don’t like snakes, especially big ones. As long as they are not poisonous or in my chicken coop I’m OK with snakes.
If you can trap the ones that are getting to your plant starts you can help yourself. A trapping program will not necessarily get them all but you can target the ones that are causing you grief. I’ve never trapped chipmunks but I’d thing a rat snap trap could be effective. A small live trap sized for squirrels could work well too. I’d probably try BOSS, Black Oil Sunflower Seeds, as bait.
It’s possible you can shoot some of them. A .22 with pellets instead of a slug can work really well and limit the range so you are less likely to hit something unintentional down range. That ammo may be hard to get. A pellet rifle can be effective. A 12 gauge shotgun could give you some satisfaction. This won’t necessarily target the ones causing your problems, but it can reduce your overall numbers.
Can you tell how they are getting in the greenhouse? Blocking their access, maybe with wire mesh, might solve that problem.
Is it possible to bend a protective cover out of wire mesh like hardware cloth to put over the individual trays of starts? Make a box to set over them.
I can sympathize, not with chipmunks but rats. I’ve had problems with rats in the garden the last couple of years. They have places to hide in the area, including under some areas I have mulched. There is an overgrown ditch/fence line just across the road. I’ve had them kill plants that were just sprouting and chew on some veggies. I know I’ll never get rid of them all, but I try to keep the numbers down. I’ve already trapped four out of my mostly dormant garden since the first of the year, just trying to keep the numbers down.