Yes -- affinization is defined (as you wrote) as the left adjoint to the inclusion of affine schemes into higher stacks. This left adjoint exists by the ($\infty$-categorical) adjoint functor theorem, since the inclusion of affines into higher stacks preserves all limits (though it certainly changes colimits). Some references for this or closely related notions: Toen's Affine Stacks (here) and Lurie's DAG VIII (available here), where the relevant notion is called "coaffine stacks". There's also a less professional and more informal discussion (in the derived context) in Section 3.2 here.