So - we had this problem a year and a half ago, it wasn’t at all on my radar that Vesuvius visit tickets would sell out. We didn’t even try to book till a couple of days before we wanted to go. I ended up getting us pricey tickets through Viator - there are probably loads of options. (I’d not used Viator before that trip but turns out it’s quite helpful for finding a lot of tour options - it’s basically like booking.com but for tours, with lots of independent providers advertising through the platform with ratings and reviews you can browse - that’s why there are loads of options that sound much the same when you search it).
I found the exact tour we went on and linking it here - no idea if it’s changed, or a different person etc, but thought it might help to have a specific tried and tested link rather than “try Viator”.
https://www.viator.com/en-GB/search/23258P144?mcid=70066 (“Mount Vesuvius Tour from Pompeii Led by an Expert Guide”).
It was pretty expensive but we were v constrained on what day / time we could go, so didn’t have many options. There are still tickets here for a couple of weeks time, and there may be other options at better value. But it was going to be a bit expensive sorting out transport up the mountainside as well (there are cheaper and more expensive options but again, last minute and constrained for timing the cheaper ones didn’t work).
I also found out that you can get last minute tickets through the official site, on the day released hourly, but that felt too stressful and uncertain with kids in tow, and also you have to get a bus or something up the mountain then try to get the tickets with poor Vesuvius wifi which I didn’t like the sound of (student / early 20s me would have been cool with that!).
Good luck!
edited to add - the tour was really excellent and I would personally recommend it - I’m a geologist and so was particularly keen to visit, and I would def not have planned to do it as a tour, but actually it was great - we met outside a nice restaurant outside the Pompeii entrance (easy to get to), it was timed so we had a good lunch there as a family before meeting up with the tour. The drive up in minivan was easy and driver and guide were informative, lots of stuff to keep teens interested, and the guide and the “expert” at the top were good at pointing out things and explaining the volcano, most of the geology was decent at any rate. The guide was also v interesting talking about wider volcanism around the Naples area.