Philosophical Zombies can have mental states and would still be Zombies. The whole point of Chalmer's Zombie thought experiment is to show that having mental states isn't enough to account for subjective/phenomenological first person experience. After all, computers have "mental states" - their internal memory states and software configurations - but don't have conscious experience.
Here's one way of looking at it: Imagine that your Zombie isn't created by evil magic, but instead is a super advanced android, one that is indistinguishable externally from a human in terms of behavior and appearance. Per Chalmers, this android has internal mental states in its robotic brain, but it still doesn't have first person subjective experience.
Another way to look at the question is historically:
The main materialist position w/r to the mind body problem in the early to mid 20th century was behaviorism. This was the idea that only external behavior was observable and talk of internal mental states was unscientific, since there was no way to measure/observe them. A problem with this view was that most accepted that the same behavior could correspond to different mental states, or vice-versa. If someone is crying, are they crying tears of sadness or tears of joy?
To solve this problem, functionalism was proposed in the 1960s, which was basically "behaviorism + internal mental states" - and was inspired in part by developments in computer science (see Hilary Putnam's work on the topic for example).
Chalmers' Philosophical Zombie concept is a direct response to functionalism: That even when taking into account internal mental states, materialism (or physicalism) still fails to account for first person subjective/phenomenological experience.