Sansa orders Ayra to execute Lord Baelish. So does this mean Ayra gets to take his face?

In theory. But they killed Littlefinger in public - and people will know that. I think they wanted people to know he was executed - that Sansa and Arya are leaders not to be messed with.

So you won't really be able to use Littlefinger's face to infiltrate anywhere. Which is the point of taking a face.