How would he do that? Not everyone has the skill to transfer chakra :lolExcept he is controlling their chakra and transfers some of his into them when needed. He can do the same by transfering their chakra into him and then using that chakra for another puppet.
Of course, you claim they have a way but logically they have no way to reproduce chakra. So you just claiming it doesn't make it so.Here we go. No, they don't have organs, but they have some way of producing chakra anyways. A finate amount would defeat the whole purpose of a Human Puppet, which is the ability to use Jutsu from the owner's life. Since he could easily use up the whole chakra reserve in a battle or two. Sasori doesn't specifically build them to contain chakra from their previous life, that much isn't possible. He creates them from their old bodies by turning them into puppets. The chakra is already there. He doesn't replenish it by himself somehow maically.
-Hearts are needed to pump and produce chakra
-As are other organs
-These puppets have none.
Honestly that explains why the hundred puppets he uses are such fodder :lol he used up their chakra to take down the nation he hyped himself with, so they just turned into normal puppets.
Wasn't the Third a Jin? He probably had a good amount of chakra without its influence and I doubt Sasori needed to use it much.