Also, as far as I understand, Sage can compute the minimal free resolution 
of 
the module of syzygies of S, and from the resolution the presentation can 
be 
assembled.


Yes. It's 
here: https://doc.sagemath.org/html/en/reference/resolutions/index.html

So it seems that the only missing bit is computation of a presentation of S.


Let phi: R[y_1,...,y_k] -> R[x_1,...,x_n] mapping by y_i -> f_i. Then, 
perhaps, your I is the kernel of phi.


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