Assuming that data (and future data to be added) are in range A2:B1000.
Assuming requested name is in K1.

=SUMPRODUCT(--($A$2:$A$1000=K1)*--($B$2:$B$1000=TODAY()-6))

or, for any range of dates, earlier in L1 and later in M1,

=SUMPRODUCT(--($A$2:$A$1000=K1)*--($B$2:$B$1000=L1))*--($B$2:$B$1000<=M1))

HTH
Kostis Vezerides