Lamination FTW
If the chair were a different type I'd have suggested solid wood (note: grain must go horizontally), but not for one like this. Laminating up a curved plywood back is the way to go here IMO as already suggested in @keshlam's Answer. Only one potential problem with this plan, and that is getting the thick veneers to do the lamination or cutting them yourself.

BTW you know these have to be glued up on a curved form, don't underestimate the amount of time making that will take.

And if you don't also make a reverse-curved one as well (female or male, depending on whether you decide to go curve-up or curve-down on the main form) you'll need a lot of clamps to make each back, and it would probably be worth using clamping cauls as well. Figure at least eight strong clamps tied up each time while the glue dries so factor that into your estimates of how long the whole process will take.

If you did want to make it from solid wood
Even though I'm a handtool guy primarily because of the relatively specialised nature of the shaping of one side of each back* this has power tools written all over it. And the main one to get this type of thing done repeatably and accurately is the router, with an appropriate jig.

Making some sort of jig to then use a router - Would still require a lot of sanding

You're underestimating the surface you can get with a router. Just generally if your bit is sharp, you use good technique (appropriate feed rate, best routing direction relative to the grain) and you take a very shallow final pass you can get nearly finish-ready surfaces straight from the router.

Plus anyway, if the surface left by the shaping operation on this project were a little rough you should always be looking at planing or scraping before sanding. Scraping is like sanding on steroids. It's faster, produces a better surface and it doesn't use up a consumable product. What are the differences between sanding and scraping?

and the jig wouldn't work all too well

If made well and used right jigs for this type of shaping can work superbly. Here are a few images showing the usual types made:

*Ideally done with a compass plane, which are rare and can be expensive.

Lamination FTW
If the chair were a different type I'd have suggested solid wood (note: grain must go horizontally), but not for one like this. Laminating up a curved plywood back is the way to go here IMO as already suggested in @keshlam's Answer. Only one potential problem with this plan, and that is getting the thick veneers to do the lamination or cutting them yourself.

BTW you know these have to be glued up on a curved form, don't underestimate the amount of time making that will take.

And if you don't also make a reverse-curved one as well (female or male, depending on whether you decide to go curve-up or curve-down on the main form) you'll need a lot of clamps to make each back, and it would probably be worth using clamping cauls as well. Figure at least eight strong clamps tied up each time while the glue dries so factor that into your estimates of how long the whole process will take.

If you did want to make it from solid wood
Even though I'm a handtool guy primarily because of the relatively specialised nature of the shaping of one side of each back* this has power tools written all over it. And the main one to get this type of thing done repeatably and accurately is the router, with an appropriate jig.

Making some sort of jig to then use a router - Would still require a lot of sanding

You're underestimating the surface you can get with a router. Just generally if your bit is sharp, you use good technique (appropriate feed rate, best routing direction relative to the grain) and you take a very shallow final pass you can get nearly finish-ready surfaces straight from the router.

Plus anyway, if the surface left by the shaping operation on this project were a little rough you should always be looking at planing or scraping before sanding. Scraping is like sanding on steroids. It's faster, produces a better surface and it doesn't use up a consumable product. What are the differences between sanding and scraping?

and the jig wouldn't work all too well

If made well and used right jigs for this type of shaping can work superbly. Here are a few images showing the usual types made:

*Ideally done with a compass plane, which are rare and can be expensive.