Carrot cake is pretty standard, but can be very variable IME. often too much cinnamon for my taste.
I've made beetroot and chocolate cake from a Green & Black's recipe book - a good, moisture chocolate cake, you couldn't tell had beetroot in. I've also had courgette cake and pumpkin scones. Pumpkin pie is also quite common.
I also had bean porridge in Borneo - red beans of some sort, and one with a greyer bean, and another with yellow, but I don't actually know what sort of beans they were - I didn't cook them, and my Malay isn't more than a few words. They were quite sweet, though. Probably quite a few pulses could work.
Root vegetables tend to contain more sugar than other vegetables (there is sugar beetroot, for one thing,) though I've not had anything with parsnip, swede or turnip... actually, I could see mashed swede could work.
Pumpkin and courgettes are both cucurbitae, like melons, so perhaps they are a bit sweeter, too.
I think you need to think about:
- texture - does it break down easily?
- flavour - although beetroot is quite an earthy flavour and you don't taste that in a cake.
- liquidity- will you need to squeeze out some liquid before adding the veg to the recipe, or add more flour or something to compensate?
I guess there could also be chemicals in some veg which could affect how well something like a cake rises, or could interact in unhelpful ways with other ingredients.
I've not seen anyone make a kale or cabbage cake, nor bell pepper (chocolate and chilli, though.) I suppose brassicas can have a tendency towards bitterness, so you might have to adjust for that.