I don't know the grammar, but I do know that C is the only natural answer. Even if you could rationalise something with grammar, that does not mean that it is something that anyone would ever say - which is the true test of whether it is correct or not.

"The government launched a scheme _____ children can have breakfast at school." (A) that (B)which (C)where (D) X I think you're suggesting that "that" of (A) can be a correct answer in addition to (C). I don't think so. The reason is because only certain nouns like "saying", "statement", "remark", etc. can have an appositive noun clause representing what is expressed. These words all mean some kind of a statement. "scheme" however is not such a word, so you can't say "a scheme that (full independent statement)". It sounds strange. You need either "where" or "in which" instead of "that" to connect "scheme" and what can be done by employing it.