Both of the following sentences are grammatical and, depending on who you ask, acceptable from the point of view of style:
✔ 1. Who is the client currently residing with?
✔ 2. With whom is the client currently residing?
First of all, it's a myth that grammarians say you shouldn't end a sentence with a preposition. It's one of those persistent proscriptions that got started by someone and never left the collective unconscious. Ask almost any grammarian, and they'll say it's perfectly fine to end a sentence with a preposition; however, it may not be the style that a particular author chooses to use when writing something.
Second, while there are traditional conventions of grammar when it comes to who versus whom, neither one is, currently, considered to actually be wrong. It's more of a preference as to which should be used. To whom (or [preposition] whom) is kind of a set phrase, so it's common to see whom used in that context. In other constructions, where whom would traditionally have been used, who is now the more common word—and it is used without comment.
So, by that token, the following would also be considered acceptable (although most people would probably use the first sentence form instead—because, currently, it sounds more idiomatic):
✔ 2b. Whom is the client currently residing with?
The following, however, would be considered awkward at the very least—if not outright ungrammatical:
✘ 3. With whom is the client currently residing with?
It looks like it's trying to get away with using a preposition at the end of the sentence by somehow constructing the start of the sentence in such a way that its use at the end is excused. Except that it's not.
In effect, what this sentence is saying is:
✘ 3b. Who is the client currently residing with with?
The second with is redundant. That one instance of with has been moved to the start of the sentence is irrelevant—there shouldn't be a repetitive use of the preposition in the first place, no matter where each is located.
If it sounds more natural to leave the preposition at the end of the sentence, then just leave it there. There's nothing wrong with doing so. If it sounds better to move it to the front, then do that.
But don't attempt some hybrid situation where it ends up being in both places at once. That's imply wrong on all counts of grammar and style.
As for which of the acceptable versions you should use, that's up to you and your audience—and the style that you decide to use in general.
Having said that, while a proposition at both the start and end of a sentence is wrong in this case, it's not true that it's always wrong.
Here are some examples of sentences that start and end with a preposition:
In what currency would you like to pay for the movie you want to go to?
On whose authority did you think I would let you in?
By what person would you like to be looked after?
They may all be phrased a bit oddly, but they are all grammatical. The prepositions being used reference different things, so they are not redundant.