It is technically possible if the extended header of the cart has it enabled then SD/NAND would be accessible.
However the main issue is that no official game ever had SD/NAND access. So no. You can not dump nand from a DSi flashcart because they would have to exploit an existing DSi game to even boot (like how the Cylco DS iEvolution works) and there are no known TWL carts that have the nand bit enabled in the SCFG_EXT section of the extended header. The header is RSA signed so you'd need the private key and the ability to re-sign the header to gain SD/NAND access from a slot-1 cart. About the only consoles you can do this right now on is dev units since you can resign stuff using dev keys. (and no, the dev keys won't work for SRLs run on retail units)