We study the existence of the random uniform attractor within the set of tempered closed bounded random sets for a family of first order stochastic non-autonomous lattice dynamical systems (LDSs) with multiplicative white noise, where the nonlinear part is an element of the hull of an almost periodic function in a suitable Banach space. Up to our knowledge it is the first time to study the existence of random uniform attractors for stochastic non-autonomous LDSs. Previously, the existence of random pullback attractors for different types of stochastic non-autonomous LDSs were investigated which are effective to describe the pullback dynamics, but, unfortunately, give no information for the forward dynamics. In fact the attraction in the random uniform attractor is uniform in time symbols from a symbol space. Moreover, a random uniform attractor is by definition pathwise pullback attracting, but has also a weak forward …