The description of the Pandava and Kaurava banners has been scattered through several Parva's of Mahabharata.
The banners of the five Pandavas and their sons is given in Chapter 23 of the Drona Parva.
On Yudhisthira's flag is a golden moon.
The standard of Yudhishthira of great energy, bearing the device of a
golden moon with planets around it, looked very beautiful.
On Bhima's is a silver lion.
And Bhimasena's standard, bearing the device of a gigantic lion in
silver with its eyes made of lapis lazuli, looked exceedingly
On Nakula's is a golden Sarabha.
For terrifying the foe, we beheld that tall and fierce standard of
Nakula, placed on his car bearing the device of a Sarabha with its
back made of gold.
On Sahadeva's is a swan.
A beautiful silver swan with bells and banner terrible to look at and
enhancing the grief of the foe, was seen on Sahadeva's standard.
On the Upa-Pandava's banners are Dharma, Maruta, Indra and Ashvin Kumara.
The standards of the five sons of Draupadi bore on them the excellent
images of Dharma, Marut, Sakra, and the twin Aswins.
On Abhimanyu's is a peacock.
On the car, O king, of the youthful Abhimanyu was an excellent
standard that bore a golden peacock, which was bright as heated gold.
On Ghatotkacha's is a vulture
On Ghatotkacha's standard, O king, a vulture shone brightly, and his
steeds also were capable of going everywhere at will, like those of
Ravana in days of yore.
Note: If I find any references to the banners of the other warriors in the Pandava camp I'll update my answer.
Now for the Kauravas. I found references to the banners of the Kaurava warriors in Chapter 17 of the Bhishma Parva and Chapter 101 of the Drona Parva.
On Bhishma's banner is a palm tree with 5 stars.
And with his large palmyra-standard decked with five stars, Bhishma,
the generalissimo of the Kuru army, 1 looked like the resplendent Sun
On Drona's banner is a sacred altar with a kamandalu on it.
The standard of Drona, the foremost of preceptors, bore the device of
a golden altar decked with a water-pot and the figure of a bow.
On Kripacharya's banner is a bull.
The preceptor of the Pandavas, that Brahmana, given to ascetic
penances, viz., Kripa the son of Gotama, had for his mark an excellent
bovine bull. That high-souled one, O king, with that bovine bull,
looked as resplendent, as the Destroyer of the three cities looks
resplendent with his bull.
On Ashvatthama's banner is a lion's tail.
Similarly, the lion-tail standard-top of Drona's son, O Bharata, we
saw, was endued with the effulgence of the rising sun Decked with
gold, floating in the breeze, possessed of the splendour of the
rainbow, the standard mark of Drona's son appeared on high, inspiring
the foremost of Kuru warriors with joy.
On Karna's banner is an elephant rope.
The standard of Adhiratha's son bore the mark of an elephant-rope made
of gold. It seemed, O king, in battle to fill the whole welkin. The
banner, adorned with gold and garlands, attached to the standard of
Karna in battle, shaken by the wind, seemed to dance upon his car.
On Duryodhana's banner is an elephant decorated with jewels.
On the standard decked with gold, of king Duryodhana, was an elephant
adorned with gems. Tinkling with the sound of a hundred bells, O king,
that standard stood upon the excellent car of that hero. And, O king,
thy son, that bull among the Kurus, looked resplendent, O monarch,
with that tall standard in battle.
On Jayadratha's banner is a silver boar.
A silver boar adorned the standard-top of the ruler of the Sindhus.
Decked with golden chains, it was of the splendour of a white crystal.
1 With that silver mark on his barrier, the ruler of the Sindhus
looked as resplendent, as Surya in days of yore in the battle between
the celestials and the Asuras.
On Salya (of Madra) banner is a goddess.
Salya, the ruler of the Madras, we saw, had on his standard-top an
image like the presiding goddess of corn, endued with beauty and
producing every seed.
On Vrishasena's banner is a peacock and a plough.
Vrishasena has a peacock made of gold and adorned with jewels and
gems. And it stood on his standard, as if in the act of crowing, and
always adorned the wan of the army. With that peacock, the car of the
high-souled Vrishasena shone, like the car, O king, of Skanda (the
celestial generalissimo) shining 'with his peacock unrivalled and
beautiful ploughshare made of gold and looking like flame of fire.
On Bhurishrava's banner is a sacrificial stake.
The standard of Somadatta's son, devoted to sacrifices, bore the sign
of the sacrificial stake. It was seen to shine like the sun or the
moon. That sacrificial stake made of gold, O king of Somadatta's son,
looked resplendent like the tall stake erected in the foremost of
sacrifices called the Rajasuya.