I have a naive question I am asking. Given a higher geometric stack X in the sense of Simpson, Toen etc is it true that there is an affinization Spec Gamma(O_X) such that Hom(X, Spec(A))= Hom(A,Gamma(O_X)) for every affine scheme Spec(A)? Or does this require some more hypotheses. I have very much a hard time finding this out.

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. 

