We have a property that has a flood zone of AE on the flood determination, but their flood policy shows X. Normally, I would request a change in coverage to match AE, but the insurance company provided us with an elevation certificate. I was under the understanding that in order to change flood zones, a LOMA is required. Am I missing something here?

It is. The only way they would not match is if the policy was grandfathered. All buildings post FIRM must be constructed above the BSE - the fact that the ground floor elevation is above the BSE has nothing to do with the zone the building is in.