There are two sets of API you can access ObjectData in Acad Map: COM API and ObjectARX MAP .NET API.
You can still use COM API as you do in VBA by setting reference to "amvbapi.tlb", and do basically the same thing as you do in VBA.
However, since you use .NET language, you'd better do it with ObjectARX MAP .NET API, by setting reference to ManagedMapAPI.dll. In the .NET API, look into namespace Autodesk.Gis.Map.Project/ObjectData/Utilities...