If fumigation kills rodents, then you want to deal with the rodents first, because you DO NOT want dozens of dead rodent carcasses stuck in your walls.
They stink SO BAD and for SO LONG.
Rat droppings are enormous compared to mouse droppings, they're little ovals about the size of raisins. Although roof rats do have smaller droppings.
You will almost never have rats and mice in the same house, mice are terrified of rats and won't feel safe living somewhere they can smell rats. However, if you have one rat, you have A LOT of rats, and they are much harder to get rid of than mice because they're about a million times smarter and more resourceful.
If the previous owners are now your renters, you can absolutely insist that they close their doors when leaving the house. This is a basic security issue. Granted, I don't know how the laws work where you live, but if you own the house, you should be able to dictate this.
Now if they'll abide by your rule? That's another matter.
If the laws allow you to, then I would start dealing with the rodents while the previous owners are still living there. A rat colony can grow in size at an astronomical rate, they are obligatory communal animals, and once one sends out a message that they have a nice, cozy, safe home, their friends will join. Rats, unlike mice, can never, ever have enough friends to live with. Unlike almost every other type of mammal, they loooooove each other and never self limit their numbers. And they can have litters every 3 weeks, so if they can't find friends, they will make them.
So yeah, figure out fast if you have rats, and deal with them ASAP.