As was explained in the comments:
The way that all the airline booking systems work, it is not possible to have people on different booking (fare) classes on the same ticket. If you have two people on the same flights in the same booking classes, it's fine. If you have two people on the same flights but different booking classes (could be business and economy, or even just flexible vs non-flexible economy), then they must be on different tickets and PNRs.
You therefore need to make two different bookings for you, one for the people in one booking class (+cabin), one for the people on the same flight but in the other booking class.
Your best bet in this case is to skip the internet, and talk to (shock horror!) a person. Be that a travel agent, or the airline's booking line, either should work. They will create the two itineraries for you, hold them while you confirm you're happy and double check that everything else is the same, then ticket the two for you. They should also be able to attach notes to indicate the bookings are linked.
Oh, and one thing to be aware of, because you'll have two bookings, if there's a problem with the flight and it gets cancelled, you will almost certainly have to speak with another human. Otherwise, you may find that you and your wife are re-routed one way, your kids another, because the automated re-booking system won't necessarily know to keep the two together because they're on different bookings.
Note that if by "mixed-class tickets" what you means was the more common "one way in economy, return in business" or something like that, then many (but not all) online flight booking systems will be able to sell you the ticket, as will the airline or a travel agent.