It is not totally out of the question! They do have Internet access, and I'm sure Donatello could fix up an eBay account; Splinter is not the most technically savvy person around, but this is important! His son's training is at stake! And also there is free shipping! I'm not sure what they'd do about a delivery address, but given the ease with which giant turtles seem to walk Manhattan (you would be amazed what a knit cap and a trenchcoat conceals) they might be able to use a PO box. A katana's the kind of thing they probably want you to sign for, though. Defeated by postal restrictions!

(Possibly one of the reasons they were so happy to befriend April was her possession of a street address. I bet tech journals and comic book shipments and books on Japanese culture started arriving almost immediately.)