It's fair to assume that water is coming from the neighbors yard but I wouldn't base any 'claim' on that unless you can prove it, or eliminate other sources such as the less-likely high water table and the clay soil scenario. Even with clay soil water shouldn't be hanging around too long because if grass and stuff is growing, water is getting through. If you could actually get footage of water flowing into your yard, or show that the grass 'blades' are bent away from the direction of any water flow, then you have something to go on.
As far as who to contact, start with Building Inspections. If it's possible, go in person to the office. Let's assume for the sake of argument that the water is coming from an 'uphill' source such as your neighbor. If it's considered natural flow, then there isn't much you or they can do UNLESS building requirements called for mitigating work on water flow at the time your neighbor's house was built. Continuing on ... if the flow IS natural AND there are no requirements, you're more than likely left with working out any solution with your neighbor, or fixing the problem on your lot. Don't go digging any trenches to channel water to another source like another neighbor. If you're gonna channel water anywhere, it should go to the street, and ultimately into a storm drain, sediment-free.
My best to ya and hope this helps. If you've got particular questions, you can e-mail me directly.