Possibility of Direct Flights from Lyon, France to Naples, Italy?

Have you checked SkyScanner?

Check Easyjet. Might be only direct flight.

EasyJet has direct flights on the schedule 2x a week in September (Tuesday and Saturday). That is as far out as their schedule shows.

Lyon's airports web page has a flight selection tool that has proven useful for me over the years. You don't have to buy through that, of course, but it's only flights to or from Lyon (almost all from LYS; the main airport) so it's useful for research at a minimum. It's the box on the left entitled Réservez votre vol (reserve your flight). You'll notice there's a box to check if you wish to limit your search only to direct flights.

You can switch from French (FR) to English (EN) up at the upper right of the web page if you wish.

Thank you all for the support. Easy Jet does have two DIRECT FLIGHTS each week from Lyon (LYS) to Naples (NAP). Once I entered dates for Tuesday or Saturday, Skyscanner also revealed the Easy Jet flight. We will definitely seek out Easyjet. The rates are very reasonable but we will have hefty fees with our 23 kg of HOLD luggage. No problem as I rather have DIRECT FLIGHT with
additional baggage costs than have to deal with multiple stops to get to Naples.

Just be sure you pay for the checked bag(s) when you buy your ticket rather than waiting till you get to the airport. Paying at the airport will make it a lot more costly.

Do pay attention to the above advice about using easyJet. Take the time to read their website thoroughly because the carrier means exactly what it says, especially concerning luggage. For instance, one carry-on bag means ONE, not a small suitcase plus a computer plus a shoulder purse. easyJet makes money from extra charges. But it encourages the customer to buy those extras in advance, on-line. The fees are much higher at the gate. Also print out the boarding pass 30 days in advance. If this sounds harsh, the airline is very efficient and good value for money. I have always been satisfied with its services.