If a child is born in another country and raised in America, can they qualify to run for presidency?

NO they would not be a native born citizen of the US.

https://www.law.cornell.edu/uscode/text/8/1401

The Constitution specifies that only a 'natural born citizen' may be president. But it doesn't explain what that term means.

Today we figure it means a citizen at birth, a person who doesn't have to be 'naturalized'. This includes anyone born in the US, and also anyone born elsewhere whose parents were American (like John McCain who was born in the Panama Canal Zone while his father was serving in the military there.)

So being born outside the US is not a deal-breaker, if the kid's parents are American. He is still a citizen from birth.

No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President; neither shall any person be eligible to that office who shall not have attained to the age of thirty-five years, and been fourteen years a resident within the United States.

They cannot, an opposite example would be Steven crowder born in America, grew up in Canada but he is now an American citizen and will be allowed to run. The rules are very clear and probably should have some exceptions but as of now you have to be born in America

Well... the president must be a natural born citizen, but I believe that a child born abroad to American parents may acquire U.S. citizenship at birth if certain statutory requirements are met

If one or both of their parents were Americans, then yes

This is why the whole birther conspiracy was basically pointless