Depends on the core. Alcan Baltek (PIC, PE) MGi (PU), Diab, You may want to try a distributor like CompositesOne or Aircraft Spruce and Specialty if you are looking for small qunatities.
Depends what you need the core for. General Plastics is a polyurethane core. They have several competitors - DUNA, Coastal, Elliot, and more. DIAB and Alcan have PVC and balsa cores. SP/Corecell has SAN cores. Rohacell are PMI cores. And there are many more - WebCore (fiber reinforced core), Plascore (plastic honeycomb), and more...