I think Frodo both succeeded and failed. Frodo was successful in that his quest was completed. He brought the ring all the way from the Shire to Mount Doom. Sure he had help, and fate (luck?) but he made the most contributions (as the predominant ring-bearer) to this collaborative act. He was successful in the majority of his quest in the least (he did bring the ring a long way). In this sense, it was neither a complete success nor a complete failure.
Frodo also failed in that he was corrupted by the ring. He vowed to himself to cast the ring into the fire. However, one could argue that even as he made this vow, he was already under the power of the ring. Did Frodo just say this because he did not want Sam to get any bright ideas about taking his ring? If this is so, then are these words even Frodo’s own? Frodo makes this vow seconds after snapping at Sam. I highly doubt Frodo was in control at this point. As an explanation to this outburst Frodo explains, “I must carry this burden to the end.” This is, disputably, an explanation for an outburst that Frodo had no control over. Is this Frodo's vow or the ring's vow?
Elrond never specifically told Frodo to get to Mount Doom on his own and personally cast the ring into the fire. It is disputable if Frodo (as himself) even made the vow to personally destroy the ring.
This entire debate rides on whether or not Frodo’s own quest was destroy the ring personally.
Because of the internal conflict between Frodo and the ring the specificity of the quest is ambiguous; thus, you could argue that he is successful in some ways yet failed himself in others. However, we really cannot be certain of either.
However, based on the context of Frodo ‘s vow to personally destroy the ring, I am more inclined to view Frodo as successful in this ambiguous quest. But still, it is impossible to be 100% certain of what Frodo’s quest specifically entails.