To amplify Brian Conrad's semi-answer, I need a more precise definition of "simply connected" at the outset. In characteristic 0 some of the classical ways of thinking about this concept can be carried over to the algebraic setting, but in prime characteristic the most common definition starts with a connected semisimple group. Over an algebraically closed field, the algebraic criterion for such a group to be simply connected is that the character group of a maximal torus be the full weight lattice.
Here the "fundamental group" of the adjoint group in the compact case is re-interpreted as the quotient of the weight lattice by the root lattice, which may also be regarded as the (scheme-theoretic) center of the simply connected group.
There may be no quotable source earlier than the 1956-58 Chevalley seminar. The classification work of Tits and others then descends to arbitrary fields of definition. In SGA 3, Expose 22 (by Demazure), Definition 4.3.3 defines "simply connected" in terms of the behavior of fibers relative to this criterion using the root datum language.