Someone in the neighborhood has a maple, and so I've got seedlings popping up around the yard constantly. When I catch them, I pull or dig them up young. But many have deep roots now, so pulling is impossible and cutting them does nothing: they just send up new shoots. Some are in extremely tight cracks along a rock wall I have, and other inaccessible places. So digging them out is not possible. What can I do?

Cut them off - that's what I do.

Spray them with Roundup. It kills the green plant only and will not poison the soil. You can also use vinegar and a saturated salt solution. Works almost as well and is cheaper. Be careful of overspray with either.

Good advice and suggestions !

I never, ever recommend Roundup without warning of the hazards. It stays in the soil for a long time and if the OP has any pets... it's more than a little dangerous.

Is the OP attempting to kill the tree or just rid his property of the roots?

Not wanting to press a point but as a chemist, we used LD 50 as the indication of toxicity. Ld50 for glysophate is 5600 Mg //kg body weight for warm blooded animals. EPA rates it's toxicity @3 on a 1-4 scale with 4 being safest. Bad on bees not warm blooded. Check Ld50 for caffeine 150 Mg /kg. Aspirin is 815 Mg /kg. Not a proponent of chemical usage for everything but In my opinion it's not "more than a little dangerous" at least on LD50 studies and that is the toxicity standard used by FDA and EPA.

I sat on my County's Pesticide Commission for 2 years about 8 years ago when applicators' licenses were a big legal argument. I am certainly not a chemist. It is very possible that test results as handed to me were slanted and/or exaggerated or just plain wrong.

I also have dogs. I do not allow pesticides on my property or in my home.

Everyone has to make a choice - allow, don't allow, restrict, something else. I am not arguing with you or contradicting your advice. I am suggesting that everyone do his/her own research and decide what is and what is not safe when there are pets on the property.

Safe is also in the eyes of the beholder (so to speak). I had a lawn spray applicator show up at my home on a cold call, trying to see his services. He said his product was so safe he would feel safe drinking it. I suggested that he do so. That was the end of the sale presentation.

No argument there but this is not a pesticide, it is an herbicide. Most long term
Pesticides have active Phosphorous and indeed are insidious.

I agree - it's an herbicide and I was wrong.

At any rate, let me put it this way - no chemicals on my property to kill weeds, grass, trees, bugs, insects...

I am not going to post all the research - which is contradictory - about Roundup. The OP could find AMHD. If he's interested he can research Roundup.

I walked around barefoot on a lawn treated with weed killer and my feet puffed up.
Imagine all our poor pets!

Garden supply places supposedly sell 20% vinegar as opposed to grocery 5%.
Or you can make it, I hear, by freezing it. Not sure what to pour off - I guess the ice of plain water in the vinegar rises, not sure. I tried it without knowing what I was doing, took the ice off, and poured what I think was concentrate on horsetail (a scary weed because it spreads quickly both by spores and runners, and I read has no chemical enemy, plus I introduced it to my area).
It killed everything, including the grass, for about a 5" radius around each weed. Not that the weed doesn't pop up elsewhere...

If the maple saplings are healthy, cutting them off at soil level could make your problem worse by growing "iceburg" tree roots that you forget about until the tops start growing again. Just pull/dig those suckers out with some elbow grease and at least try to hack out most of the root- if they are too large to pull out they will likely survive being cut back back anyway. And remember an ounce of prevention is worth a pound of cure when it comes to weeding. When the tree saplings are starting to take root and are still tiny, just take a stirrup hoe to the area, and spread bark mulch thickly to prevent seeding in the first place.