[Question] Can you dump ds ROMs to an original ds game cartridge?

no, you cannot

 

There are several methods to dumping DS games. With the DS, your options are rather limited
1: Dump using a Slot2 cart like EZ-Flash VI. This requires either slot1 to boot into DS-Mode or FlashME
2: Wifi dump, which is PAINFULLY SLOW. Requires a flashcart.

With the 3DS, you can use a hacked 3DS and dump games using GodMode9

 

I think he wants to write a rom to a ds cartridge.

 

I was thinking that too, but I was really hoping it was typo or misunderstanding.
If that is the case, then it's not possible.

To the OP: DS games are Read-Only, you can't rewrite them.