This is a multi-part answer, as your question is a bit vaguely worded.
To give you a quick summary, here's what I think is most likely:
- Arya has the skillset of the Faceless Men, but is not a Faceless Man.
- The Many-Faced God allows Arya to change face, because she is doing things that the MFG likes (even if the MFG is not the reason why Arya kills who she kills)
You can find a more elaborate response below.
1. Arya is a Faceless Man
What do you mean when you say that? As I see it, there are two possible meanings:
- Arya is an active member of the order of Faceless Men.
- Arya has the skillset of a Faceless Man.
1 is false. She leaves the Faceless Men and retains the identity of Arya Stark. However, it is true that she was invited to become a Faceless Man (evidenced by Jaqen saying that "a girl is finally no one"), but Arya declined.
2 is true, as evidenced by her killing Walder Frey in disguise and later explicitly stating that she has the skill when talking to Sansa.
2. Why can Arya use the faces if she's not a member of the order?
I think this is what your question boils down to.
The skill of facechanging seems to be twofold. There is a physical aspect to it, Arya needs to cut someone's actual face off in order to wear that face. This physical aspect is a skill that Arya was taught. Just because she leaves the Faceless Men does not mean she has unlearned the skill.
As @BCDotWeb said in the comments:
If you've studied to be a cook, do you afterwards ask the school you've studied at for permission whenever you make a sandwich?
Arya doesn't need permission to cut off faces. It's not a patented procedure.
But then there is also a magical aspect to it. When Arya puts a face on, how does this change her body? Why does her voice, posture change (and, according to the books, why does she have the memories of that person?)
This is never truly explained (as no magic has really been explained yet in the show). We can only trust in the words of Jaqen, who said that the Many-Faced God gives them this power.
In other words, Arya can change face because the Many-Faced God has granted that power
As I see it, there are two possibilities as to how it works. Note that these are of course more speculation than anything else, but I'm simply trying to list every logical possibility.
1. The Many-Faced God has automated the process.
By this I mean that the MFG doesn't need to "approve" every face change on a case by case basis.
We could prove this theory by seeing what happens if someone else steals one of Arya's faces and put it on. If it changes them, then that means anyone can change face (which then explains why Arya is capable of changing face even though she's not a member of the order).
Either way, it's likely that the face itself needs to be cut in a specific way, which points back to a skill that Arya herself has learned.
2. The process is not automated, the Many-Faced God approves of Arya changing face every time she does so.
In other words, the MFG does need to "approve" every face change on a case by case basis.
We could prove this theory by seeing what happens if someone else steals one of Arya's faces and put it on. If it does not change them, then that means that only certain people are allowed to change face.
Initially, this makes no sense. Arya is not a member of the order of Faceless Men, why would the MFG allow her to change face?
Unless the MFG approves of Arya's plan. What if he is aware of Arya's future (other gods have shown to be similarly prophetic), knows what Arya is going to do, and approves of that?
After all, he is the god of Death, and Arya intends to kill people.
So while she is not a Faceless Man, that doesn't mean that the MFG can't still approve of her actions regardless.
No one has said that the MFG gives favors to only the Faceless Men. (pun not intended)
Similarly, you could wonder why the Lord of Light resurrected Jon, who has not become a devout worshipper (comparatively, Beric has become a believer through his resurrection). The simplest explanation is that Jon was resurrected for a good reason regardless of what Jon's personal beliefs are. The Lord of Light doesn't need Jon to preach, he needs Jon to do the thing he is destined to do.
And if you agree on the example of Jon, then the same can be true of Arya and the Many-Faced God.