Try calling a lumber yard that isn't part of a national chain. Tell them you want 5/8" MDF. If they tell you they don't have it in stock, tell them you want to special order it. You might also try
WoodFinder to locate a lumber yard in your area that can special order the stuff. WoodFinder doesn't have a search for MDF, so try "Hardwood-veneer plywood". Places that carry hardwood-veneer plywood will tend to sell to custom cabinet shops, so they're more likely to stock or at least special order 5/8" sheet goods. Not everyone is listed in WoodFinder, so be sure to use your local yellow pages, too!
If you can't find 5/8" MDF and 5/8" sheet good can work: plywood, OSB (assuming you're going to laminate it), melamine, particle board, etc.