Does salt water conduct mostly by the ions travelling through the solution, or by electrons collectively flowing or hopping through the solution like in metals?
Let's look at pure Water with pure Sodium Chloride in it. We know that in Sodium Chloride, the Sodium atoms give one electron to the Chlorine. This results in both atoms having closed shells.
For electrons to collectively flow/hop through a material, you need to have electron states that are delocalized over many atoms. This is what happens in metals: The individual atomic orbitals overlap and form bands. Now, band theory also tells you that for conduction to occur, you need a band that is partially filled. If all your bands are completely empty or completely full, you cannot have conduction.
Since, as I have described at the top, your Sodium and Chloride ions have closed shells, there will be no transport of electrons among the atoms. If you think about it in terms of hopping: Both Sodium and Chloride will be very happy with the number of electrons they currently possess. Hence, they will be very reluctant to either give up or accept hopping electrons. I will not go as far as claiming that these processes are entirely unlikely, just strongly suppressed.
So yes, it'll be the ions itself doing the (bulk) of the transport.
EDIT: My reasoning above rules out that the electrons will behave the same as in a metal, i.e. forming a Fermi gas. As pointed out in the comments, chemical solutions exist where electrons are free to move around, behaving like ions. Another answer explains why this cannot happen in water.