I kind of assumed that different people were "assigned" to "Naruto duty" back when he was a toddler. It's just no one stuck around, because no one wanted to be around the "demon child". They all just immediately requested transfers whenever they got that job, until Naruto was old enough to look after himself. Everyone knew, they just all never talked about it, because they all sort of held a grudge against the Nine-Tails for being responsible for killing one of their leaders. Old grudges and learning letting go of the past being a big theme of Naruto and all.
...I mean, that doesn't make a whole lot of sense. You would think SOMEONE would have taken baby Naruto in, if only because he was the Fourth's child, and all. You'd probably get some sort of stipend for that kind of thing from the government. I guess the idea was that their hatred of the Nine-Tails kind of blinded them to that. But, at some point, I guess you just have to shrug and let that plot hole go. You would think they'd use one of their dozens of spin-offs to clarify what happened there, but as far as I know, they never did.
Jiraiya couldn't have done it, since he was wandering around for his "research" projects. The Third Hokage probably SHOULD have been doing it, but he held a grudge against the baby, for hosting the demon that killed his protege. So, I guess some nameless series of faceless ninja probably were responsible for taking care of baby Naruto. Like those low-rank missions where Naruto had to go find lost cats. Only it's taking care of this orphan baby no one wants to be around, lest he explode in demonic fire suddenly, I guess.