Several caveats in my answer to that. Firstly, I am not sure if your jurisdiction uses the same code that my area does. Second, the code edition that I am most familiar with was in effect 10 or so years ago. I have read the current code cover to cover, and in most ways it is very similar, but I can't quote wording from memory, so....
Third, I am not looking at your specific situation, so I have to make some assumptions, as follows. I am assuming that there is a vent in the wall behind your lavatory, that the trap arm drains into that stack, and that from there the vent either goes out the roof or revents back into another vent stack. And, that IF the lav stack is a wet vent for the rest of the bathroom, that it is 2" or larger at least to the point that the lav drains into it.
You would need to remove the vanity cabinet. Cut out the wall behind it to expose the drain arm and the stack. If the stack is between the existing lavatory and the proposed lavatory, replace the existing sanitary tee with a sanitary cross, 1 1/2" each way. If the stack is to one side or the other of either the existing or the new, replace the 2"x1 1/2" sanitary tee with a 2" sanitary tee. If between, route 1 1/2" drain pipe to the approximate center both lavatories. If to one side, use 2" to the first drain, then use a 2"x1 1/2"x1 1/2" sanitary tee on it's side to stub out for that drain, then use 1 1/2" to the second lavatory. While you are at it, run the water piping over to the new lavatory as well.
Now, if you have to penetrate any of the studs with 2" pipe, even if you drill the proper size hole, and even in a nominal 2"x6' wall, you will have to double that stud. In a 2"x4" nominal wall, you will have to double the studs that you drill through even if the pipe is 1 1/2". And if you have to double up the studs, you can do that to no more than two consecutive studs.
Off of the top of my head, and following the code that I know best, that is what is required to bring this up to code. MOST inspectors that I have dealt with would probably OK the wye under the existing lav, because they are sentient, logical creatures. Some inspectors aren't.