I wouldn't keep them with axies mostly because if the axie manages to get it in its mouth it'll be a biiiig issue. I do however have tiny snails that occasionally infest my tank from a plant I hadn't quarantined well enough. They will disappear for a while then show up again. I usually try and get them out and put them in my fish tank to avoid any run ins with my axolotl.
I definitely agree with Hayley, just too risky. It will cause a lot of problems if/when they swallow them. If you are looking for algae buddies you can try ghost shrimp. They have a hard time seeing the shrimp and make the occasional snack when they do get ahold of them.
Generally it has the risk of impaction, I have read of some people who have kept large mystery snails with their axolotls and had no problem so maybe that could work out, I'd definitely avoid Malaysian Trumpet Snails, Nerite Snails, or any snails that could potentially be eaten. Mystery snails might, emphasis on the might, be okay, just know that your axolotl could try to eat the snail and die of impaction. I would make sure and have the largest Mystery snails possible because if it can fit in your axolotl's mouth they will try to eat it.
Snails constituted part of a wild axolotls diet, they pose minimal risk as long as hard shelled species and trapdoor species are avoided. I have raised thousands of axolotl with thousands of snails with zero problems , for the axolotls atleast bit of attempted genocide on the snails though. Axolotl have evolved to eat snails.
If anyone would like to post that captive axolotl are now a domesticated species so far removed from their wild counterpart that they cant tolerate a wild diet, please take a deep breath, slap yourself in the face , go stand in a corner and contemplate the error of your ways young man/lady/other. Repeat as often as required.