That is a GREAT question.
The first place I would look is for lockers at the train station in Rome where you will arrive when you take the train to Rome from Civitavecchia.
If you plan right, that will be the same train station where you end your tour. Then you can go back there, get your bags and get on a train to take you to the airport.
Otherwise - the only other option I can suggest is that you find a quality hotel and ask the concierge if they will watch the bags for the day. Expect to tip them pretty well (10 Euro?)
Or it is possible to hop-on tour provider will have lockers or someone to watch them, but my personal experience with these tours in Rome is that they were not helpful.