Can the US president be impeached when citizens collect enough signatures on a petition?
There is no binding petition process on federal level in the United States.
The impeachment process according to article 2, section 4 of the constitution can only be initiated by congress. They are only allowed to do so "for Treason, Bribery, or other high Crimes and Misdemeanors". But congress also define what "high Crimes and Misdemeanors" are in that context (the last time it was lying about cheating on his wife), so they basically start the process when they feel like it.