You may find it's less the health risks of unwashed fleece and more the gunk on their wheels.
Because of ours being a mass participation event, we reluctantly decided to use washed fleece. So although we waited for the sheep to be hand clipped, in fact we used a similar fleece that had already been clipped, washed and dried. Personally I'd rather use it straight from the sheep.
You might be able to arrange to use only drum carders that are used for greasy fleece, and similarly hand carders. Otherwise, here's how to clean up the drum and hand carders afterwards. You'll need some washed, spare fleece. Wet some with meths and card it until the tines are all sparkly clean. Then card some more of the washed fleece, without any meths, to clean off the meths!
Meths will likewise clean any grease off wheels afterwards. And again, finish with a clean dry cloth to remove any remaining meths.
Places on wheels to specifically clean are of course the path the yarn takes, so orifice and hooks, but also the flyer shaft and bearings. And for the pernickety who don't usually spin in the grease, also the drive wheel bearings.
I find a bit of grease in the fleece helps to keep the wood fed, so I guess that means that there may be grease will want wiping off (using meths) of any lacquered surfaces.
Do let us know how you get on, Sally! After our escapade, a competitive one has kind of got to be our next challenge...