These are some egg-free, dairy-free desserts I have tried and been satisfied with. Note that they are not all vegan, but wherever milk is listed as an ingredient I use almond milk, I use coconut oil instead of butter, and my "egg" of choice is 1 Tablespoon of flax meal plus 3 Tablespoons of water (mix together and let sit for 5 minutes).
No Bake Walnut Cookie Bars - These are pretty rich but they do taste good.
Eggless Vanilla Cupcakes - I made these for AK's birthday last year so C could have some too. I made the following substitutions: subbed coconut oil for the butter, almond milk for regular milk, honey for the sugar, and used regular whole wheat flour instead of cake flour. Baked at 325 for longer than recommended (maybe about 20 minutes) and they turned out great!
Strawberry Beet Cupcakes - I know. But you must try them before you judge. I made them for C's birthday and they were a hit with everyone there. And yes, there were lots of people there.
Pumpkin Bars - I doubled the recipe and had to cook longer than it says, but they turned out really well.