(Sorry about the few sources, but this what I've come up with so far and if you think it is insufficient I can do more research.)
From what I gathered it seems like you could go either way and either both decisions would and wouldn't make sense historically mainly because cannons were not used commonly as siege weapons but rather against soldiers, ironically they were more likely to be mounted on castles then shot at them.
But as far as "could" any clan have gained accesses to cannons it seems likely because closer towards the end of 16th century gunsmiths in Japan were catching up to Europeans and even in some cases were more advanced (e.g. some matchlocks crafted in Japan were designed in a way that allowed them to fire in the rain, such designs re unknown in Europe during the period).
And Japanese cannon technology improved greatly from around the 1570s with cannons shooting a shot of about 10oz to 1610s where they shot weighed around 18lbs..
If you believe the cannons in STW2 are somewhere in the middle of this with destructive power then I would say it would be justified to have them.
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