Not if Beijing has anything to say about it, and they do. In 2007, the Chinese Government enacted a law that asserted the Chinese government was the final authority in the process that recognizes a reincarnated lama claiming that "all Tibetan spiritual leaders, including the Dalai Lama, must be approved by the Chinese central government."
The basis of China’s claim is that it possess the Golden Urn, a relic that was used by the Manchu emperors to anoint Tibetan Lamas. The “Historical Procedure" is based on the use of the 17th Century Golden Urn system whereby names of proposed reincarnated lamas would be drawn from an urn and sent to the Chinese Government for the final decision.
It is the same urn that China used to appoint the 11th Panchen Lama, rejecting the boy identified by the Dalai Lama as the reincarnation of the 10th Panchen Lama.
On a separate occasion the Dalai Lama has indicated that the Tibetan people collectively should have a role in choosing his successor. Speaking at a gathering of world religious leaders in Amritsar, India, he discussed the prospect of a referendum before he dies:
"When my physical condition becomes weak, and there are serious preparations for death, then the (referendum) should happen", he said.
So, it’s really not altogether up to Dr Lobsang Sangay, or the CTA or even the Dalai Lama.