The AugPAKE spec says it can be implemented over elliptic curves. This sounds very promising, but they don't actually back that claim. Can this really be achieved? If so, how would one go about mapping the existing protocol to elliptic curves?
Yes for sure you can do that. Mapping this protocol to an elliptic curve setting is just like mapping DH key exchange to ECDH key exchange. In AugPAKE you work in a prime order $q$ subgroup of $Z_p^*$ and in the EC setting you use a prime order $q$ elliptic curve group.
Observe that in the EC setting a multiplication of group elements in AugPAKE is then point addition and exponentiation in AugPAKE is then scalar multiplication.