Is this gap at the base of my wall allowing water to enter?
That's not the wall, exactly, but the stucco finish on the wall. I think I see the brick or block sub-structure behind it.

If that's the case, the stucco is not supposed to go into the ground. It's an above-grade finish, like most siding would be. Drainage away from the building is handled by proper grading (slope, surface material) and rain gutters, among other things. 

Since this is apparently a rental, and since filling that gap probably won't solve the issue, you need to bring it up to the property owner. If they won't address the problem, investigate your legal recourse or consider finding alternative habitation.