Aside from the fact that your field/control name is a Reserved Word, the problem you are having is because the current record has not been saved yet. Moving off of the current record saves it.
Code:
Private Sub Delete_Account_Click()
On Error GoTo Err_Delete_Account_Click
Me.Delete = "Delete"
DoCmd.RunCommand acCmdSaveRecord
CurrentDb.Execute "AppendFromUnixUserID", dbFailOnError
CurrentDb.Execute "DeleteFromUnixUserID", dbFailOnError
Exit_Delete_Account_Click:
Exit Sub
Err_Delete_Account_Click:
MsgBox Err.Description
Resume Exit_Delete_Account_Click
End Sub