So I'm making a hill dwarf underdark druid who became a druid because while exploring he came across the mother of all giant spiders that for some reason gave him a long look and let him live. Renouncing all claim on his family's businesses and position in society he disappeared into the underdark. The question is with no wood and a special focus on spiders would it be acceptable to make a wand out of a spider leg. I figure it's natural and the chitinous nature of it could make it suitable. I'd allow it but I'm very new to the game and DMing. I also get super excited for random sideways thinking like this
Druidic Focus. A druidic focus might be a sprig of mistletoe or holly, a wand or scepter made of yew or another special wood, a staff drawn whole out of a living tree, or a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.
A wand made from a spider's leg sounds a lot like "a totem object incorporating [body parts] from sacred animals", but it's perhaps up to DM discretion.
I'd assume most DMs should allow it, regardless of whether they might consider the spider "sacred" to your druid, because it is a very specific item and shouldn't upset the balance of the game in any way–you're still able to have it taken from you, you need to hold it to cast spells, and you can't just craft one from at will.
This is entirely DM purview.
If you are the DM these types of logical and consistent changes that fit your playstyle and world-building strategy make perfect sense. There are guidelines in the DMG starting on page 235 in Chapter 3 on making changes like this although they are a bit vague the general idea is the same.
The base rules are just that, base, and the DM can make changes and is encouraged to do so to improve story-telling and game play and contribute to the overall fun of your games.
Don't be afraid to make such changes but I would caution you to take them incrementally and only make large changes when you get some experience under your DM's cap.
Another option that comes to mind is zurkhwood, which is a fibrous fungus that is found in the Underdark. They detail it in the Out of the Abyss adventure; a wand made of such material could be wrapped in spider's silk and whatnot to give it flair.