A case can be made for all three versions. To start with, if needed is not necessarily a parenthetical—it can be, but it can also just be a constituent in the sentence. If you pause before it, it’s a parenthetical; if you don’t, it’s not.
If we treat if needed as a parenthetical, then (B) is definitely the correct option: parentheticals need enclosing commas; that’s our cue that they’re parentheticals. The conjunction and also needs a preceding comma. So:
(B) Here's a summary of our discussion, and, if needed, more detail can be found at www.website.com. √
The comma before and would be considered obligatory to most purists, since and starts a separate main clause here; but informally, you can probably get away with omitting it:
Here's a summary of our discussion and, if needed, more detail can be found at www.website.com. (?)
If we don’t treat if needed as a parenthetical, then there is no need to put a comma before it. It is still a kind of adverbial-like constituent that starts the clause, however, and those are commonly, but not mandatorily, separated from the rest of the clause by a comma, especially if they’re very long:
Yesterday(,) I went swimming.
Every night at around half past one when I’ve been tossing and turning in bed for an hour without being able to get to sleep, [← pretty much has to be there] I get up and drink a glass of water.
If needed is quite short, so it can take a comma or not, depending on preference. Thus, neither (A) nor (C) can be considered incorrect:
(A) Here's a summary of our discussion, and if needed, more detail can be found at www.website.com. (√)
(C) Here's a summary of our discussion, and if needed more detail can be found at www.website.com. (√)
So essentially, all three are correct, and even a fourth one would probably not land you in big trouble. Remember that punctuation has nothing to do with grammar, although it can (and should) make text more understandable. That is its purpose. In the end, punctuation is really a matter of style.
A different matter is whether you really want to be using that and in this case at all. The two main clauses are quite independent of each other and fairly orthogonal in meaning: there is no logical progression from one to the other that really justifies an and to begin with. A better alternative may be to simply split the clauses into separate sentences, separated by either a full stop or a semicolon:
Here's a summary of our discussion. If needed, more detail can be found at www.website.com.
Here's a summary of our discussion; if needed, more detail can be found at www.website.com.