No, and it is arguably prohibited by the Constitution. First, it specifies quite clearly what the punishment for impeachment can include.
Judgment in Cases of Impeachment shall not extend further than to
removal from Office, and disqualification to hold and enjoy any Office
of honor, Trust or Profit under the United States; but the Party
convicted shall nevertheless be liable and subject to Indictment,
Trial, Judgment and Punishment, according to Law.
This all clearly applies only to the person convicted.
As noted in the comments, Congress does not generally have the power to impose punishments, except when the Constitution says otherwise:
No Bill of Attainder or ex post facto Law will be passed.
A law punishing anyone other than the impeached official could be considered as an unconstitutional Bill of Attainder.
Second, although not directly applicable to impeachment, there is another clause that would likely be taken into account in any dispute:
The Congress shall have power to declare the punishment of treason,
but no attainder of treason shall work corruption of blood, or
forfeiture except during the life of the person attainted.
Corruption of blood is the punishment of the heirs of someone for their actions, preventing them from inheriting titles or estates. While this does not have direct bearing on impeachment, if Congress tried to impose a penalty on the children or relatives of an impeached official in defiance of the previous clause, the judiciary would probably consider this clause to offer some insight into the constitutional position of such an action.
As a side note, you have mentioned some of the advantages of corruption of blood, but consider also the disadvantages: a person is barred completely from certain occupations because of the crimes that another person committed, which could include young children who never knew their family or adults who have no contact with them.