The vice president becomes the new President if the President vacates his office for any reason, including death, resignation, or forced removal via the impeachment process.
Note that the President would remain president after impeachment until and unless the Senate votes to convict. Being impeached is roughly equivalent to when a criminal is indicted or charged with a crime. After that a trial is held and unless they are convicted, nothing happens. Both Andrew Johnson and William Jefferson Clinton were impeached. Neither was convicted by the Senate so they each remained president.
If the Vice President is impeached as well, the Speaker of the House becomes President.