Train from Genoa to Menton FR via Ventimiglia - can’t book!?

No need to book. The segment is covered by regional trains (one change at Ventimiglia). There are some intercity trains from Genoa to Ventimiglia, but no need to book especially because you don’t know when you will actually get to the station as flights are sometimes delayed. Just buy the ticket when you get to the station. There are no discounts on Regional trains for advance purchase and the IC would offer only insignificant discounts for that segment. There is no risk of regional trains selling out all seats. They work like subways, seats are on a first come first served basis. If all seats are taken (very rare if even possible), you just stand up until a seats frees up.
You can also purchase tickets on the Trenitalia app (but only to Ventimiglia) but don’t do so until you are on the bus or taxi on the way to the station.

When is this?

Always tell us when, when asking help with train travel. Always.

Genoa Airport has actually a train station, connected by a shuttle bus to the terminal. You can book that on Trenitalia:

So you can book in Trenitalia "Genova Aeroporto C. Colombo" to Ventimiglia. But I would not book that now. Buy that ticket using the Trenitalia app once you have arrived. (There might also be ticket vending machines at the airport, but I cannot verify that).
Regional tickets do not need to be bought in advance, and you cannot buy them far in advance anyway, also buying via resellers does not always work too, as Trenitalia uses a rather unique way of delivering electronic tickets for regional trains.

Ventimiglia - Menton you can just buy in Ventimiglia, or using the SNCF app.

Thank you Roberto and Wengen!
The options are surprisingly confusing. Even being sure which service is the official one helps!
I am traveling on Wednesday afternoon in July.
The apps seem to be the way to go - including buying the shuttle ticket from the airport.
I was trying to nail it down in advance because the train schedules are quite tight.
But if I have the shuttle booked in advance then I should be able to keep moving and being ok everything else en route.
Thanks again for the help!

You do not need to worry about the schedule being tight. This is mass transit., If you miss a connection you take the next. So you don't need to book in advance.

The "shuttle" to the train station is just a bus. And if you buy a ticket from the airpot to Ventimiglia you can just hop on the next bus to the airport train station. They run every 15 minutes. You are not limited to the bus you selected. The train to Ventimiglia is just a train. Trenitalia does require you to "check in" for a train using the app, so if you end up taking a different one than the none you selected during purchase you need to change that in the app.

This is relatively trivial. It is like arriving at Chicago and taking the L in to town. How far ahead would you book that?

Just to add to the above.
In Genova there are multiple train stations. There are two main ones, called Genova Piazza Principe, and, a bit more to the east, Genova Brignole.
In addition to those there are several secondary stations.
The one very close to the airport, referred to by Wengen, is called Genova Sestri Ponente. The AirLink shuttle bus can take you to that station in 5 minutes. Since that station is not one of the principal stations, InterCity trains (IC) don’t stop there, to catch those to Ventimiglia you need to go to the Genova Piazza Principe station, which is linked to the airport via the Volabus shuttle bus. That bus ride to Piazza Principe is a bit longer (maybe 30 minutes).
But if you take a cheaper Regionale train (R) or Regionale Veloce (RV) train, those trains also stop at Genova Sestri Ponente station near the airport.
Personally I wouldn’t bother with the InterCity, I would take a Regionale. It’s cheaper, and only marginally slower (maybe 20 minutes longer to Ventimiglia, since it makes more stops).
As Wengen says, a Trenitalia Regionale train is mass transit, like the L in Chicago (just much cleaner, more modern, nicer and faster than the L or any NYC subway.

Italiarail is just a US ticket agent, which also uses other names. Raileurope used to be the international sales department of SNCF but was sold off and is now an independent agent based in Paris.

The cause of this problem may be something to do with new rules for using italian regional ticket sold online - which are likely to cause a whole lot of trouble for agents, who may simply decide not to get involved.

But as said, there is no point to booking in advance for regional trains so donlt worry about it - as long as you have double checked th itinery and know where you are going..

Thanks, All for the help.
I took the Regionale from Sestri Ponente.
Here’s what I learned:
Regionale means “local” essentially. I guess that is “within a region” rather than between which would be the Intercity.
The shuttle to the nearby station Sestri Ponente, is super quick and there is one machine on board that takes a tapped card for the 2 or 3 Euro ride.
I did see the sign for the Velobus outside the airport and would have taken that because it said it goes to the train, but the shuttle came first and when I asked the driver he said “Yes get in!”.
No one was in the Genoa airport to ask and no airport WiFi - a very small operation.
The Regionale train was very well equipped with screens indicating upcoming stops with times, so we could figure out where to transfer.

Was there no way to buy the through ticket from the airport to Ventimiglia (that includes the bus)? I would have been interested in knowing how that worked.

I think there was in the app. (You have to enter the airport as your starting point.) But I would have been confused about which bus to take without knowing which station the train was leaving from and which bus to take with that ticket. I was doing the naive traveler approach. I probably would have taken the other bus if it came first, to be honest. And the longer bus ride would have been compensated by the quicker train from the main station. And maybe the conductor would cross-honor my ticket if I had ended up on the wrong path to the same destination.
But I went with pay-as-you go. The only drawback there is if there is no time to buy a ticket before boarding.
The bus to the main station is called Velo. The bus to the local (sestri ponente) is just a shuttle (not sure how they named it) - like a city bus.