I think that there is two levels of answer in this question, whether we talk about an exact scheme (the RG is one in principle), or about the practical implementation/calculation.
If one could implement the RG scheme exactly, one would capture emergence, since this is equivalent to solving the problem exactly. So, if you know the correct question, that is, if you know what is the correlation function you need to look at to observe this emergence, then this exact implementation of the RG will give you the correct answer.
However, the correct observables might be very different from the one you would guess from your microscopic theory, and it might be very complicated to find what is the correlation function you need to look at.
Related to that is the practical implementation of the RG, where you expect some non-trivial flow (most probably non-perturbative) that you might not be able to take into account. Then you must guess what the emergent degrees of freedom are, write a QFT for them, and start your RG all over again.
However, one should keep in mind that there are non-perturbative implementations of the RG that can do most of what is described here, at least for some specific problem. The perturbative RG is not all that there is! Two examples.
1- The classical XY model in 2D. It is well known that the dominant contribution to the physics are given by the (un)-biding of the vortices, which are very unlocal objects. The usual approach is to start from your spin Hamiltonian / field theory, do the transformation to the vortices degrees of freedom, which then give a theory which looks like a 2D coulomb gas, and then do the pertubative RG from there. However, by implementing a non-perturbative RG (NPRG) scheme, which is described only by the Hamiltonian of the spins, one can describe most of the qualitative and quantitative feature of the XY model without ever including the vortices by hand! See for example the review arXiv:0005.122, as well as arXiv:1004.3651.
2- In the case of a quantum model, one can describe the Bose-Mott transition in the Bose-Hubbard model using symmetries argument to find what are the two universality classes of the transition. One then introduce a low energy action with some unknown effective parameter to describe the low-energy physics close to the critical point. One cannot perturbatively start from the microscopic action to compute the effective action because the system is on the lattice and strongly coupled. However, using the NPRG, one can effectively do this calculation and explicitly find the low-energy degrees of freedom as well as the effective parameters starting from the microscopic action, see for example arXiv:1107.1314. Nevertheless, I should tell that in this case, this is not a strong form of emergence (the low-energy degrees of freedom are not very exotic, the fixed points are well known, etc.), but still, one can do a lot with the RG!