Is the following sentence grammatically correct?
With the exception of a few members of my family, you are the only three people to whom I actually care what happens.
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Irrespectively of the fact that the use of "whom" (rather than "who... to") might per se be judged somewhat over-formal for the sentence in question, I think your sentence has another issue which makes it sound odd. To try and explain things informally, your sentence essentially has the problem that you're trying to make [to whom] "jump too far" from its underlying/logical position in the sentence.
By "too far", we are actually referring to a structural limitation rather than the number of words per se. To explain this, consider sentences such as these:
1a. [Jane] is feeling [better].
1b. I wonder whether [Jane] is feeling [better].
2a. [The jars] are stored [dangerously].
2b. This is the cupboard where [the jars] are stored [dangerously].
Now, imagine taking the bracketed phrases and making them into the subject of a question:
1a. [How] is Jane feeling?
1a. [Who] is feeling better?
1b. *[Who] do you wonder whether is feeling better?
1b. *[How] do you wonder whether Jane is feeling?
2a. [What] are stored dangerously?
2a. [How] are the jars stored?
2b. *[What] is this the cupboard where are stored dangerously?
2b. *[How] is this the cupboard where the jars are stored?
In the (a) sentences, the transformation is fine. But in the (b) sentences, although we essentially apply the same logic to transform the statements into questions, the results are odd or ungrammatical.
It turns out that in general, the elements represented by a WH-word or phrase like "who...", "how...", "what ..." etc like to move to the "nearest place" that they theoretically could (in posh syntactical speak, this is sometimes called the "shortest movement principle"). And in the (b) versions, that "nearest place" that they would like to move to is essentially the place where there is already another "WH" word in the way. For example, in (2b), the words 'what' or 'how' would really "like" to go in the place where the word 'where' is; moving to the start of the sentence is a "longer" movement, because there is a potential place for them to move to that is nearer (just that the word 'where' is in the way).
So in your sentence, what is essentially happening is that the word 'what' is behaving a bit like 'where' or 'whether' in these examples: the preferred place for the phrase 'to whom' to move to would really have been where 'what' is, so it is forced to move further up the sentence, and the result is odd-sounding.
(P.S. This is a slightly Chomskyan view in terms of where phrases "like to move": you could also see things in terms of how we are used to interpreting sentences.)
1) "you are the only three people about whom I care" - most formal in style 2) "you are the only three people whom I care about" - formal in style 3) "you are the only three people who I care about" - neutral in style, but some old school grammars insist that "who", being a subject pronoun*, would be incorrect here. 4) "you are the only three people I care about" - who[m] omitted**, informal in style but correct, and the most common form, especially in spoken communication.
The 'to' is inappropriate. You should use about or you could even use nothing.
you are the only three people about whom I care what happens.
you are the only three people whom I care what happens.