– Have you received the letter which I sent today using my private e-mail address?
Should the adverb "today" be placed at the end of the sentence? If not, could you please explain why? Is the adverb not attached at the level of the verb phrase (VP)?
The reason I ask, though, is because of Jackendoff's take on the placement of VP-adverbs (a summary of which is included below). Unless I have missed it, the above position is not mentioned as a possibility, and this has left me wondering as to its grammaticality.
Hello, MlleSim—Please note that the end of a clause is not necessarily the same thing as the end of a sentence. In your sentence, you have three clauses: the main clause ("Have you received the letter"), a relative clause ("which I sent today") and a nonfinite participial clause ("using my private e-mail address").
The adverb(ial) "today" occupies clause-final position within the relative clause, "which I sent today." Prior to wh-movement, the clause reads "I sent which today." The adverb(ial) "today" comes at the end of the verb phrase, and the end of that verb phrase is the end of that clause (the relative clause).

In the attachment, "clause-final" position is listed as a possibility for VP adverbs.