We demonstrate coupling between the atomic spin- and orbital-angular momentum (OAM) of the atom's center-of-mass motion in a Bose-Einstein condensate (BEC). The coupling is induced by Raman-dressing lasers with a Laguerre-Gaussian beam and creates coreless vortices in an F=1 ^{87}Rb spinor BEC. We observe correlations between spin and OAM in the dressed state and characterize the spin texture; the result is in good agreement with the theory. In the presence of the Raman field, our dressed state is stable for 0.1 s or longer, and it decays due to collision-induced relaxation. As we turn off the Raman beams, the vortex cores in the bare spin |m_{F}=1⟩ and |-1⟩ split. These spin-OAM coupled systems with the Raman-dressing approach have great potential for exploring new topological textures and quantum states.