Spider carcass is obtained as a drop from jungle spiders or sarachnis. These carcasses are used for cooking spiders on shaft and spiders on stick.
The carcass can be attached to skewer sticks obtained from cutting thatching spars with a machete, and can then be cooked into a spider on stick. Meanwhile, spiders on shaft can be cooked by attaching the carcass to arrow shafts. Cooking one of these spiders on a stick or shaft is a requirement for the medium tasks in the Karamja Diary. This requires a Cooking level of 16. The shafts are destroyed when spider is eaten, while the skewer sticks are reusable.
To obtain spider carcasses, it is recommended that players travel to Yanille. There are a number of spiders on the peninsula due east of the town; players may run back to the Yanille bank to store their carcasses. Players can access the peninsula quickly through the fairy ring code CLS, which teleports players next to Hazelmere's house. Note that these spiders (level 44) do not appear on the minimap as yellow dots, so players of a lower level should take caution because although these spiders are not poisonous, they are still aggressive.
For an exhaustive list of all known sources for this item, see here