Do we ever find out if Elayne takes the test to become a full Aes Sedia? We know that she was raised by Egwene, and that Nynaeve was also, but that Nynaeve took the test for it to be official. Do we have a record of if or when Elayne also takes the test?
As of the end of the series, no.
The following exchange in Towers of Midnight tells us that Elayne wasn't going to take the test while pregnant:
"I need both of you to return to the Tower," Egwene said. "Elayne, I know what you're going to say - and yes, I realise that you are Queen, and that Andor's needs must be met. But so long as you haven't taken the oaths, other Aes Sedai will think you undeserving."
"True," Elayne said. "But the timing is ... awkward. I don't know if I want to risk swearing the oaths while pregnant. It might harm the children."
And she's still pregnant by the end of the series, so it's a safe assumption that she's never taken the test by then. Whether she takes it after the end of the series is another question, to which there is no canon answer.
Thanks to ryan, Creation Edge, and Mike Edenfield for a little assistance with spoilers :-)
To expand upon @Randal'Thor's answer, we not only don't see Elayne test for the shawl, but:
We see the Amyrlin Seat dead, and no one left to run the White Tower, and no one to enforce Elayne taking the test
In extra sources, such as The Wheel of Time Companion and interviews, we're not given any information about how that is handled. However, I have my doubts it will necessarily happen, because
Part of the reason Elayne needed to take the test is because she, like Nynaeve, was a friend of Egwene's and was raised when Egwene was only the rebel Amyrlin. Those political reasons disappear because of the aforementioned death.
If the Amyrlin decides it's necessary to push the Queen of Andor and mother of children of the dragon under the new structure of the White Tower, then it'll probably happen, eventually, but it's impossible to say what the Amyrlin will want.