The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general groups over a regular ring of dimension $\leq 2.$ But since reductive groups are especially nice, maybe they do have such a representation? If not all reductive groups do, which of them do? 



Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations?
In Corollary 3.2 of the paper


R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), 16–34,


this was proved for semisimple group schemes or, more generally, for reductive group schemes which are either split reductive, or semisimple, or with isotrivial radical and coradical, or over a normal base S.


