For a compact quantum group G of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L-infinity(G) into an ultrapower of the hyperfinite II1-factor (the Connes embedding property for L-infinity(G)). We establish a connection between the Connes embedding property for L-infinity(G) and the structure of certain quantum subgroups of G and use this to prove that the II1-factors L-infinity(O-N(+)) and L-infinity(U-N(+)) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N >= 4. As an application, we deduce that the free entropy dimension of the standard generators of L-infinity(O-N(+)) equals 1 for all N >= 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O-N(+).