Sturtevant, The Pronunciation of Greek and Latin, says of the change in pronunciation of C before front vowels (p. 167): "The epigraphical evidence of this change is not abundant enough to inspire confidence before the sixth century". He doesn't discuss the evidence any further, unfortunately.
Note that the change is a little more complicated than just "C became [ts]". The outcomes of Latin C before front vowels in the Romance languages are varied: e.g. French and Portuguese have [s], Italian and Romanian [tʃ] (the sound in "church"), Peninsular Spanish has [θ] (the sound in "thin"), while Sardinian retains [k]. It seems plausible that [ts] was an intermediate stage on the way to the French and Iberian outcomes.
Something similar happened to G before the same vowels: it first became [j], with later specific developments in the daughter languages. Sturtevant thinks the change to [j] occurred by AD 500, so it seems likely that the C change would have taken place at about the same time.