Actually, Wilson (who received the Nobel for his work on renormalisation -- though that's not to say he was the first one to think of it) worked on numerical renormalisation. The starting point is to formulate (in an abstract and somewhat formal (i.e. not very computable) way) the Exact Renormalisation Group Equation.
There are various ways to formulate this, one choice is to either fix bare action and integrating out the higher frequencies to get an effective action, or to fix the desired action and take the cut-off to infinity; the other choice is in the form of the cut-off --- one might want nice properties such as the regulator obeying the symmetries of the action (Poincaré, gauge, etc.).
Up to this point, everything is exact, and in the words of a lecturer of mine, "therefore useless". The other problem is then to solve said equation, which is actually a set of uncountably infinitely many coupled non-linear differential equations (or even differentio-integral equations). The solution, numerically or otherwise, is then a separate problem of approximating suitably. On this front there is not a clear answer, and is the main focus of research. One may easily implement the kinds of analytic approximations that one finds in textbooks (e.g. $n$-th loop, $n$-th order in perturbation, etc.) but there is a lack of real understanding of how the various choices affect the outcome, and when they are appropriate for a given problem.