The scriptures do refer to God as vengeful several times. Here are just a few I think are most significant (All translations from the Tanakh 1985):
See, then, that I, I am He; There is no god beside Me.
I deal death and give life; I wounded and I will heal: None can deliver from My hand.
Lo, I raise My hand to heaven And say: As I live forever,
When I whet My flashing blade And My hand lays hold on judgment,
Vengeance will I wreak on My foes, Will I deal to those who reject Me.
I will make My arrows drunk with blood -- As My sword devours flesh --
Blood of the slain and the captive From the long-haired enemy chiefs.
O nations, acclaim His people!
For He'll avenge the blood of His servants,
Wreak vengeance on His foes, And cleanse the land of His people. (Deuteronomy 32:39-43)
Assuredly, thus said the LORD:
I am going to uphold your cause And take vengeance for you;
I will dry up her sea And make her fountain run dry.
Babylon shall become rubble, A den for jackals,
An object of horror and hissing, Without inhabitant.
Like lions, they roar together, They growl like lion cubs.
When they are heated, I will set out their drink And get them drunk,
that they may become hilarious And then sleep an endless sleep,
Never to awake -- declares the LORD. (Jeremiah 51:36-39)
Thus said the Lord GOD:
Because Edom acted vengefully against the House of Judah and incurred guilt by wreaking revenge upon it --
assuredly, thus said the Lord GOD:
I will stretch out My hand against Edom and cut off from it man and beast,
and I will lay it in ruins; from Tema to Dedan they shall fall by the sword.
I will wreak My vengeance on Edom through My people Israel,
and they shall take action against Edom in accordance with My blazing anger;
and they shall know My vengeance -- declares the Lord GOD.
Thus said the Lord GOD:
Because the Philistines, in their ancient hatred,
acted vengefully, and with utter scorn sought revenge and destruction --
assuredly, thus said the Lord GOD:
I will stretch out My hand against the Philistines
and cut off the Cherethites and wipe out the last survivors of the seacoast.
I will wreak frightful vengeance upon them by furious punishment;
and when I inflict My vengeance upon them, they shall know that I am the LORD. (Ezekiel 25:12-17)
God sometimes acts to avenge his people because of the close affinity is feels for them. And though God also judges his people for their sin, as far as I could tell, the scriptures never use the word 'vengeance' to describe those judgements.