Here is a few responses some time ago to a similar question that might help -
Do it yourself solution - get your utilities located to avoid hitting them, then start at the wet spot and work your way uphill, digging holes with an small diameter auger or posthole digger (or power auger if ambitious and have 2 strong people to do it), to see if you can trace the source. If you do hit water in a hole, let it fill as much as it will (typically a minute or two), then muddy the water a bit and see which edge it clears along first - that is likely uphill, and the direction to move to trace the flow towards its source.
You can start shallow in the wet zone and may have to go deeper as you work uphill, and if water is coming under the house you may have to start with several holes a few feet from the house on the uphill side to find where it is coming in toward the house as it likely spreads out around the house. Look for differences in grass - if getting water in a dryer environment will be lusher, if getting too much water will be yellowing.
Or start along the neighboring property line to see if you can pick up soggy zone, or area where water runs into the holes. This is where I would start if digging is pretty easy - though you don't really know how deep to go to look for the water, like you would if working uphill towards the source. Start with holes spaced about 20' apart along the fenceline, then if you hit no difference in dampness split space that to tighter spacing. This method would result in less holes, but the problem is if water is deep you might miss it with shallow holes, though I would guess if you front yard is saturated, probably running pretty shallow - maybe even just a shovel hole depth under the lawn.
Professional solution - pay a drilling company (make sure they get utility locates marked before drilling) to come in with a small power drill and drill holes to track it. Pricey - likely about $200/hour and could take several hours. Cheaper alternative - pay a fence company to come with their fence hole auger and drill a bunch of 4' holes for you to inspect and trace with at maybe $100/hour.
If you do hit water in a hole, let it fill about as much as it will (typically a minute or less), then muddy the water a bit and see which edge it clears along first - that is likely uphill, and the direction to move to trace the flow towards its source.
I water IS coming from their yard and you are unable to get them to shut it off, two solutions - take legal action against them (assuming it is articially caused flooding, not natural runoff), or put a french drain or swale along property line or uphill of house to intercept seepage and divert it to a save discharge area.