Early arrival in Paris

There is a luggage storage called City Locker. We used the one just south of Notre Dame by a block or two . You pay for a locker (and they are big) online before-hand and are given a code for access. We used it in these same circumstances a few years ago and it was very straightforward.
Here's the address for the one we used
6 Rue des Bernardins, 75005 Paris, France

Google "City Locker Paris" and you'll see a google map with the different locations if this one is not best for you.

And there are other versions. Just Google "Luggage storage Paris" to see them (NannyBag, BagBnB, etc)

 
This list of railway stations with storage appeared here recently: Gare du Nord, Gare de l’Est, Gare Montparnasse, Gare d’Austerlitz, and Gare de Lyon. Also suggested were these businesses: Holibag, Nannybag, Eelway, and City Locker, as suggested above. Its link is http://www.citylocker.fr/site/files/index.php

Another suggestion was: https://www.francetoday.com/travel/paris/city-locker_the_first_automatic_left_luggage_facility_in_central_paris/

(I have tried none of them myself; it's just research.)

Do these 'city lockers' have security screening of luggage before they are stored? Or are they the next terrorist disaster waiting to happen?

Regarding lockers between France and Germany, there is a difference. Germany has coin lockers, France discontinued that in the mid-1990s.

The lockers at Nord and Est require your luggage to be screened, then they are stored. I've used it Est.

My 2 star hotel at Nord allows me to store the luggage there regardless if I have checked out or hours prior to check-in., eg, if I show up at 6 am when the room, obviously used the night before, is not ready, and am told that the room won't be available until 2 pm, I can easily leave the luggage there...very accommodating, no need to use the lockers at Nord.

Nothing waiting to happen.

Try citystasher. You can use the map to find a place to store your bag close to your VRBO.

Fred the OP is not using a hotel so can't use that advice. And while the train stations screen luggage before accepting it, it isn't clear what private locker companies do. I have no idea if 'city locker' screens luggage; surely someone who has used them can tell us if the luggage is searched or X-rayed before acceptance as it is with the train stations.

City Locker certainly fulfill a need, as the frequent requests for information on luggage storage for the early arriver clearly shows. My only problem would be the price they charge of €15 per luggage.

If your arrival is after the current renter is supposed to leave, the owner may allow you to store your luggage in the apartment until the cleaners have finished. We have done this several times.

Regarding lockers between France and Germany, there is a difference.
Germany has coin lockers, France discontinued that in the mid-1990s.

If you use the lockers in the major train stations I would still be prepared with coins just in case. In theory they accept coins and bank cards, but when I was at Gare du Nord last September and storing my bags, they were only accepting coins as payment (I had to go upstairs and buy a €2 coffee with a €20 bill and then sheepishly ask for all the change in coins -- the friendly but resigned cashier had clearly had to do this for quite a few individuals before me).

Try asking your VRBO manager for a suggestion. They may have had this question before and be able to provide info. on the good (or bad) experiences of their previous guests with storage facilities. (Hopefully, they might allow you to drop the luggage earlier than your actual check-in time.). I'm thinking that at least some of your extra time will be eaten up with going through the customs process. I don't like to leave my luggage anywhere and would probably just figure out a way to deal with it until check-in time, but I know others feel differently.