I guess both food processors and blenders would act like big blade grinders. Blade grinders give particles of a wide range of sizes, from fine dust to chunks. This gives uneven flavor extraction (if you can taste it), and requires the use of the finest filter. If I had to choose, a small food processor might be superior, since the blades on blenders are pitched to stir more than to cut.
Lacking a burr grinder, which gives more consistent particle size and therefore predictable extraction, I'd repurpose an old pepper grinder!
If you have to choose between those two evils, use the food processor. It is made for reducing solid food to crumbs. A blender is something entirely else - it is meant for turning mushy food into a uniform liquid. While you can try throwing dry food into a blender, you 1) get a worse result, and 2) risk damaging the blender.
I will also note that neither of them is the right tool for the job. If you try the food processor and the results are bad, that's not because the blender would have been better, it's because to get it done well, you need a grinder.
You can't beat a burr grinder as they produce consistent sized grinds.
But, if you're trying to produce the best cup of coffee with either a food processor or a blender. Post processing the grinds would probably be your best bet to avoid an inconsistent coffee grind size with an in uneven flavor extraction.
With no experience using the food processor or blender as a coffee grinder I'd choose the blender as a blender is going to stir and chop resulting in more even grind size. (based on the blade slightly slanted orientation in a coffee grinder. is no slant on a food processor)