A little searching turned up:
Ring epimorphisms and C(X) by Michael Barr, W.D. Burgess and R. Raphael (article).
They consider this question for rings of the form of continuous functions on a topological space. They quote the following characterisation of epimorphisms in the category of commutative rings:
Proposition: A homomorphism f : A → B is an epimorphism if and only if there exist matrices C, D, E of sizes 1 × n, n × n, and n × 1 respectively, where (i) C and E have entries in B, (ii) D has entries in f(A), (iii) the entries of CD and of DE are elements of f(A) and (iv) b = CDE. (Such a triple is called a zig-zag for b.)
This seems a little more complicated than localisation, though I haven't checked the details.
They then go on to prove that
2.12: A subspace Y of a perfectly normal first countable space X induces an epimorphism if and only if it is locally closed.
If I understand all the terminology correctly, then this implies that
C([0,1],ℝ) →C((0,1),ℝ)
is an epimorphism.
There are plenty more references in that article, and it would be nice to have an actual zig-zag for this situation. But in the spirit of open-source mathematics, I thought I'd post this and see if someone (possibly me later on) can fill in the details.