There is no specific provision in Article 9 that bars aircraft carriers. The "spirit" of Article 9 is that the JSDF is not to maintain "offensive weapons", which islike saying "no scary guns, only cute guns". It's an effectively meaningless limitation.
Currently, the JMaritimeSDF fields 2 Izumo class "helicopter carriers". In theory, they are different from true carriers for fixed wing aircraft, in reality, there's no real line in the sand distinction. A big helicopter carrier is a small true carrier, and the Izumo class is the largest helicopter carrier in the world:
With minor modifications, the Izumo can be converted for operating short take off and landing fixed wing aircraft like the F-35 lightning. Propeller-driven drone shouldn't be a problem either.
With more dramatic modifications, a ski ramp can be installed and more traditional fixed wing jets could potentially be operated as well. But this is only in theory (though I would not be at all surprised if the designers had such a modification in mind).
And of course, these are not the only helicopter carriers that the JMSDF fields. However, the older models are too small to even contemplate operating fixed wing aircraft, other than VTOL designs like the Harrier, which Japan has none of.