It would make sense but I do not know for sure. Are you going to get enough rain during the summer to support barrels? I know you could fill the barrel with a hose but then it makes more sense to go direct to house water.
That's a good question. I thought of just using the rain barrel to supplement, and running the drip system off the spigot. That way I could even use a timer if I wanted.
One other thought on rain barrels is that most are filled by collecting rain from a roof. I would not feed plants I'm eating with that water as it may pick up chemicals from the roofing material. Not scientific, just opinion.