Hi Alisa, welcome to Permies!
My partner is vegan and occasionally struggles with facial acne. I asked her for her thoughts on this, and while she doesn't use any kind of acne product, she's found that dietary changes, charcoal soap and obsessive bedding changes were the most effective protocol for her.
Cutting down/out her sugar intake was the largest factor- not just sucrose but the sugars and sugar alcohols so common in processed foods (like most if not all vegan substitutes for things!). Additionally, cutting back on carb intake (not very easy for vegans, I know) is extremely helpful.
There's evidence that carbohydrate rich foods (bread, pasta, potato, etc) can trigger acne breakouts. Also increasing her intake of healthy fats (avocados & coconut oil) helped a lot.
Most charcoal soaps (often branded as 'black african soap') are vegan, and the charcoal does a good job of cleaning out pores without completely stripping your face or damaging your skin.
Finally, changing bedding. A pillowcase is a cloth bag you shove your face into for 6-10 hours a day. Naturally things build up on it, and can irritate your skin. Article about that
here. Also worth mentioning, (it's in the article as well), our detergent was causing bedding to irritate her skin, which contributed to the problem. Making our laundry detergent has helped with that.
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