If I remember correctly, the Vanish series (R and XC) have the least padding, something like 14mm front and rear. It's still more padded than I'd like, coming from some of the Merrells, but I liked the weight and fit enough to switch. I run in the Vanish XCs.

I've never seen them in a store to try on though, so you'll have to order them from Altra's site and send back the ones you don't like. I find they run a little small.

All Altra shoes are zero-drop, so heel and forefoot are at the same level. It doesn't mean there's no padding.