The Ultimate Star Wars factbook indicates that his disfigurement occurred during an altercation with a bounty hunter.
Hideously scarred by a bounty hunter, Evazan has death sentences in 12 systems hanging over him
The Star Wars Character Encyclopedia - Updated and Expanded edition factbook notes that it was during this attack that he met and befriended Ponda Baba
Evazan is a smuggler and murderer with many enemies across the galaxy. A bounty hunter once tried to destroy Evazan, scarring his face. An Aqualish trouble-maker named Ponda Baba saves his life and became his partner in crime.
EU / Legends.
His injury was the result of a (glancing) blaster burn from bounty hunter Jodo Kast.
"Evazan was wheezing badly, and as he reached the bay, he turned and fired at me. My armor easily deflected the bolt, and I moved up with confidence.
"It was then that I realized that he was cornered. He had entered the wrong docking bay or something. He had no place to run and it was only a matter of time before the better man won.
"I moved up, doorway to doorway, trash bin to trash bin, until I was at the edge of the bay. Evazan was hiding behind a ship, a typical beat-up Corellian light freighter.
"Moving up into the bay, I started laying down covering fire. Luck was with me, as a stray bolt grazed his face. Just the way I like them. Not dead, just damaged a bit.
Star Wars - Galaxy Guide 1: A New Hope