This answer should probably come from someone not so far removed from child-rearing days; however, it seems pretty simple to me:
1. Make it an assignment communicated very clearly to them so they'll understand that taking care of the puppy all the time is their responsibility, they will be held accountable and there will be penalties if they fail to do it.
2. I don't think a rotation system of one child doing it for one day, then switching off to the next, etc. will work, so perhaps they should do it together every time--feeding, watering, cleaning up after the pup, taking it out to potty, etc.
3. The "penalties" for not doing their puppy-care assignment can be such things as: loss of part of their allowance for each failure to do what they're supposed to do; loss of TV/video games/phone privileges for successive failures; not allowing them to go places or do things with friends (what used to be called "grounding" back in the day); loss of treats that they would enjoy otherwise.
4. Children should be held accountable for their responsibilities. If they begged and begged for a puppy with promises ("We'll take care of it, Mom...we promise!"), they should have to do so--no letting them off the hook except in emergencies.
5. If these things don't work, give the puppy away to a good home with someone you know will take good care of it (not a shelter) and tell the children they can't have another one until they're responsible enough to take care of it. At this point, they may beg for a second chance, but if you've been through multiple instances where they didn't take care of the puppy, I think second chances should wait until they're more mature. Puppies aren't toys that can be tossed aside until the children are ready to play with them. They're live creatures that need care, attention and love. Some children may not be ready for the responsibility until they're older.