Not with cellulose, and likely not from the ground up. Space elevators are typically imagined as being bootstrapped from a starter cable lowered from orbit. And the tensile strength for a safe one is an order of magnitude or two higher than good steel, to say nothing of plant fibers.

Could we build a space elevator with artificial silk?

https://www.slu.se/en/ew-news/2017/1/spinning-spider-silk-is...