Hi! Just checking, did you happen to click on Hardmode? That would permanently disable the Tanning Rack and prevent you from getting Leather. You can confirm by clicking on the cow, if it just moos happily, that means Hardmode is active. Unfortunately, the only way to turn it off is to reset your game.
Tanning Rack should be unlocked at depth 90 (after Sand zone), there will be 2 choices: build the Tanning Rack (standard mode), or skip building Tanning Rack (hardmode).