you can buy fresh cane stalks that will be shipped in from overseas (typically) for you to juice on-site. It's popular with the flannel/moustache/whole foods crowd. In fairness, fresh cane juice is good. Anyway, you'll need a mill and it's a mess (trust me, we do this every day from cane we grow). You can buy sugar cane syrup on amazon, grown in the south.
It wouldn't be rum, as mentioned, but fresh pressed sweet sorghum and sweet sorghum syrup both make really good spirits. Had a blast doing some back at Barrel House in Lexington, ky several years ago.