Giverny

Maybe you can take the train to Vernon and store your luggage. Then do your Giverny thing and follow that by picking up your luggage and getting on the train to Rouen. The Cafe Tabac de la Gare in Vernon used to do luggage storage back in the day. Then they became part of the Holibags family of businesses that held luggage but I think Holibag has gone out of of business. You could call the Cafe and see if they still store bags for visitors -- maybe on another platform other than Holibag. Otherwise, if you can't find any luggage storage just take the train to Vernon from Rouen one day. It's backtracking but it's like an hour all total of backtracking so in the scheme of things not a big huge deal. One thing to point out is that if you are able to store your luggage in Vernon, you will likely miss the shuttle bus that coordinates with the trains to get you to Giverny. Not to worry, there are taxis in Vernon but I am giving you the heads up that you might have an additional cost.

I would rent a car near Giverny, then, unless you were not planning to rent one until much later!
Not possible on a Sunday, but you have options both in Vernon (Hertz and Avis, but not by the train station) and in Mantes la Jolie (Sixt, Europcar and Hertz right by the station, would be my choice, also the train to Mantes la Jolie is more frequent and "free" if you have a valid Passe Navigo for your time in Paris).

I saw some old reports of luggage storage in Vernon, otherwise, but have no firsthand experience and don't remember anything recent.

Lastly, yes you would backtrack in your original plan, but it's not that bad...

Will you have been in Paris a few days or is this right from the airport? Asking because Giverny is a nice day trip from Paris.

This would be a trip on our way out of Paris. We want to soak up as much time as we can while in the city before heading away. I did a google search for luggage storage (didn't know that was a thing) but it seems no longer available in Vernon. Bummer. Thanks for the tips.

We are planning on doing the same and haven’t found the solution yet either.

It is so interesting to me that so many say stop in Giverny on the way to Rouen and rent a car there but there is no luggage storage in Vernon!

Elizabeth, the car can act as luggage storage in Giverny. There is a dedicated car park, just back up so that the trunk can't be easily opened or seen, take the valuables, and I would feel safe enough!

True but I want to continue by train to Rouen and not get a rental car until a couple days later, not get a car there just for luggage storage.

I just Googled luggage storage Vernon France and got this encouraging hit:

https://www.eelway.com/en/luggage-storage/vernon/gare-de-vernon-giverny

It seems hopeful, but I entered the information and it seems like Vernon no longer has luggage storage.

Think there are local hotels that would offer it?