Contrary to popular belief, federal officials can be impeached even after leaving office. This situation already happened with Secretary of War William Belknap in 1876. He was impeached after resigning his office, and the senate held a trial but did not convict. U.S. Senate: War Secretary's Impeachment Trial
Theoretically the same sort of thing could happen to Joe Biden or others who have held office in previous administrations. Impeachment and conviction result in “disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States”, which effectively would ban him from becoming President or serving in future administrations even if he is not holding office at the time.
As far as a basis for impeaching Biden consider his bragging about getting the Ukrainian prosecutor fired who happened to be investigating his son. ( For example see: Joe Biden Brags about getting Ukranian Prosecutor Fired - YouTube)
Democrats now claim that actual criminal activity is not required to impeach and convict; perceived “abuse of power” is sufficient grounds. Of course, the likelihood of Democrats in the House allowing an investigation of Biden is remote, but Biden could still face eventual impeachment based on precedents that Democrats establish with their treatment of Trump.