No. John is not a reincarnation of Elijah or anyone else. No, Christians do not accept reincarnation. Here is the scriptural basis for my answer ...
This was John’s testimony when the Jewish leaders sent priests and Temple assistants from Jerusalem to ask John, “Who are you?” He came right out and said, “I am not the Messiah.” “Well then, who are you?” they asked. “Are you Elijah?” “No,” he replied. “Are you the Prophet we are expecting?” “No.” “Then who are you? We need an answer for those who sent us. What do you have to say about yourself?” John replied in the words of the prophet Isaiah: “I am a voice shouting in the wilderness, ‘Clear the way for the Lord ’s coming!’” Then the Pharisees who had been sent asked him, “If you aren’t the Messiah or Elijah or the Prophet, what right do you have to baptize?” John told them, “I baptize with water, but right here in the crowd is someone you do not recognize. Though his ministry follows mine, I’m not even worthy to be his slave and untie the straps of his sandal.” This encounter took place in Bethany, an area east of the Jordan River, where John was baptizing.
John 1:19-28 NLT
And just as each person is destined to die once and after that comes judgment, so also Christ was offered once for all time as a sacrifice to take away the sins of many people. He will come again, not to deal with our sins, but to bring salvation to all who are eagerly waiting for him.
Hebrews 9:27-28 NLT