A headless Fender jazz can certainly be made - with or without headstock.
Without headstock.
So you want to chop off the h/s.... OK. Rickett points out how the truss-rod gets in the way if it is positioned at the H/S. If it does, you have to leave enough headstock to fit the headpiece.
The only type of headpiece that will fit is the Hipshot or Steinberger type and they both accept normal strings. Since it will be positioned further away from the nut, you won't be able to use double ball end strings.
But you have to figure out how to deal with the h/p and h/s angles. You'll see from pictures that those are fitted at an angle very different from that of a Fender headstock - or what will be left of it. I can't offer a solution here. It looks like a bit of a nightmare.
The bridge. DavidCertain makes a sensible point about tuners sticking out of the body not being desirable. I say, that's the best way to defeat one of the headless main reason to exist - tuning reliability. Plain silly. (IMHO

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You are better off carving the body to achieve some sort of protection for the tuners and space for your fingers to tune. Remember to make sure bridge and neck alignment is OK.
Lastly, I think you need to ponder over the more... shall we say? philosophical side of the whole thing

The bass you're working on wasn't designed as a headless. Both ergonomics and looks will be (badly) affected. Good design solves both practical and aesthetic problems with inspired simplicity ( i-pod anyone?). Poor design doesn't.
I have seen decapitated Fender Jazz and from a pure designer's point of view their looked unresolved and amateurish at best.... IMVVHO

Not a surprise you don't see them around.
Of course you have to like it and I say, if you do then go right ahead. Just make sure you do like it.
The other option you've got is to leave the headstock and convert it to a string retainer function. It can be done. At the least, I've done it...er...just about to finish it

(hey, Rickett, thank you for remembering that!).
To my eye it is a pleasing solution which harmonizes all the best of a headless system with the gusty, timeless look of a Fender jazz. I had to make my own string anchors together with many other mods. If you like you'll find it in this forum - look for 'Building the Über-Jay'.
Lastly, after having played live both headless and traditional basses for years, I recently wrote my thoughts on this controversial matter on my blog. You're welcome to take a look if you like
>>here.
long winded certainly....but I hope this helps. ciao
