Is there a way to compute the norm form of a number ring?

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The easiest way is probably to write the linear factor that the norm form splits off over the number field and compute its norm (which is the norm form) as a resultant taken with the minimal polynomial of the field generator:

sage: M.<a>=NumberField(x^3-x-8)
sage: B=[M(a) for a in M.maximal_order().basis()]
sage: R.<x0,x1,x2,a>=QQ[]
sage: f=sum([R.gen(i)*B[i].lift()(a) for i in [0,1,2]])
sage: f.resultant(a^3-a-8,a)
x0^3 + x0^2*x1 - 6*x0*x1^2 + 8*x1^3 + 2*x0^2*x2 - 11*x0*x1*x2 + 44*x1^2*x2 + x0*x2^2 + 92*x1*x2^2 + 64*x2^3

You should generally try to avoid working over splitting fields. It's rarely necessary and, for larger degree extensions, often infeasible.