How to make a mob vulnerable to only one kind of weapon.

i dont know if its possable but you could make it take no damage overall then add a procedure checking if the player attacked it

and had "wooden shovel" in mainhand then deal damage but i dont think this would work with sharpness if for some reason you had that on the shovel

if ya run into more issues you can post a screenshot of the procedure and i can check it out

Create a procedure "On entity attacked" and check if the attacked entity is a Crab, and if yes check if the item in main-hand of the attacking entity is a Wooden Shovel. If not, cancel event that triggered global trigger.

It'll work with Sharpness and other damage changing effects such as Strentgh.