Am a dual nationality Australian/Japanese travelling to Europe this summer on a tourist visa. Is it possible to apply for a working holiday visa either while in Europe?
It depends on the specific country, even countries that are part of the Schengen area haven't harmonised their policy in this area. For example, you can do that in Germany (using either your Japanese or your Australian passport) or in the Netherlands (only for Australians, Japanese citizens are not covered there) but not in France. There is in fact a Working Holiday Visa program in France for Japanese and Australian citizens, so you could stay there as well but it's necessary to apply for the right visa from outside France in a country where you are considered a resident.
I don't really know the rules for Italy but this document from the consulate in Melbourne and this page from the embassy in Wellington (rules could in principle be different for New Zealand citizens but I assume they reflect some general Italian policy) both suggest that applying in advance is necessary for this country as well. There are also significant restrictions on the type and amount of work you could do compared, e.g., to the Dutch program.
Note that you could still visit France and Italy for a few weeks in the summer (your passports allow visa-free short stays), then go to a European country where you can get a WHV, work there for some time and then come back to France or elsewhere in the Schengen area for another visit. But you cannot stay for one full year in either France or Italy if you haven't arranged a WHV before leaving Australia (or wherever you reside now). See Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries?
I am not 100% positive about this but it could even be possible to apply for the French WHV from Germany or the Netherlands if you are staying there on another WHV. You simply cannot do it from France. So you could first get a WHV in one of the easiest country to bootstrap the process and then move to other European countries as needed (on top of the effort to figure out the rules and complete the paperwork, there are fees and delays for each application so this only really makes sense if you intend to travel in Europe for a longer time).