No, you'd need an addition device called the "ezy 3 in 1" flash cart that fits into the gba slot, thus enabling the r4 to boot games. Lack of ram, etc is why the r4 can't on it's own.
With using a DS Flashcart on its own, No. With using a Slot-2 Device called "EZ Flash V 3 in 1" and a DS Flashcart at the same time, Yes. The EZ Flash V 3 in 1 is sold seperately and can be found from most places that sell DS Flashcarts. Beware that only DS Lite consoles are capable of using it. So forget about it if you use a Fat DS, DSi or DSi XL.
The Fat DS can use this?: I know it may be possible with modifiying its casing, but I'm really trying to make this as simple as possible for the person asking for help.