Yes, INDULGENCE is a Scrabble word!

Letter values in Scrabble US: I 1, N 1, D 2, U 1, L 1, G 2, E 1, N 1, C 3, E 1.

Is it a valid word?

INDULGENCE is a valid word in the official Scrabble dictionary (US), the UK Scrabble Dictionary and Words with Friends (ENABLE).

an inability to resist the gratification of whims and desires
a disposition to yield to the wishes of someone
the act of indulging or gratifying a desire
foolish or senseless behavior
the remission by the pope of the temporal punishment in purgatory that is still due for sins even after absolution

Also: self-indulgence.