This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space H^3. Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada-Watanabe theorem. This leads to the existence of a local strong pathwise solution.