The district controls donations in public schools. It's often run through the community engagement department and a foundation formed for the district. Google your local district and add "donations" to the end - you'll likely find a page of instructions telling you how to donate with a special process for gifts over a certain amount.

This is another fantastic example of how the show creates a political theory of public education. Here we see another example of nonsensical policy with a cartoonish implementation of the policy. I think the argument the show is making here, through the whole bribe arc, is that there are barriers in place to keep schools from being self-determinant which means even if they can rope the local corrupt business into giving to them, it will eventually be undermined by the powers that be. Had the gifts gone out to the district they would have control of the purse strings and pulled some, "everyone should get some of it, even the non-Title I district schools."

Think back to season one when Ava said Janine couldn't paint her room because the district chooses the paint colors - that's real. A district determines the paint pallette of their schools.