Preparing as much as possible of a cake in advance
If you look at how boxed cake mixes do it, you'll get the right idea. They combine the dry ingredients and you have to add the wet ingredients. There are a couple of good reasons for this: spoilage (not too much of a problem if you're taking about a couple of days in the fridge); and gluten formation, which requires water and will give a tough, bready texture.

I suggest you mix the dry ingredients, and measure each of the wet ingredients into its own container (combining the apple sauce, vinegar and vanilla should be fine). Use containers that you can get everything out of or your measurements will be off. Then you don't need to refrigerate the oil or dry mix but do need to chill the wet mix and presumably the almond milk.

The buttercream should be fine made in advance - it certainly would be if it was based on dairy butter and milk. Of course this needs chilling. It also needs to be applied to a cool cake so it doesn't melt.

The other option is to bake the cake in advance and only ice it when you get there, but some recipes keep better than others. This has the advantage that the cake will definitely have cooled. I'm no expert on the keeping properties of vegan cakes in general, let alone this recipe, but many of the vegan cakes I've eaten have been the sort of thing that would keep their texture for a few days.