Ask Mr. Smarty Plants

The Lady Bird Johnson Wildflower Center makes no recommendations for nor against chemical products for destroying unwanted plants. What we are totally prepared to comment on is the undesirable invasive and non-native Ficus pumila, which is native to temperate and tropical Asia. The Wildflower Center does, indeed, make recommendations against the growth or propagation of any plant not native to North America and the area in which the plant is being grown, as well as any plant, native or non-native, that becomes invasive and detrimental to other plants. 

So, we are in total agreement with you on the subject of getting rid of the fig ivy, but we are not lawyers, so we have no idea what the legality of destroying someone else's plant is; it is not in our realm of expertise. We don't like the use of the word "spray" when dealing with herbicides. Sprayed herbicides make no differentiation between native or non-native, invasive or welcome, they just kill.

For some more detailed information on herbicide for this plant, please read this article from plantanswers.com The "friendly plant killers of choice". Then, we recommend you read the comments, especially the negative comments, from Dave's Garden Ficus pumila. You will note that many of the negative comments are from California. Note also this comment from the same website:

"Danger:
Seed is poisonous if ingested
Parts of plant are poisonous if ingested
All parts of plant are poisonous if ingested
Handling plant may cause skin irritation or allergic reaction"

Now, what to do? First, if you are on friendly enough terms with your neighbor, recommend these same articles for their reading. Next, there is no way it can be illegal for you to kill plants on your own property. It seems unlikely, considering the nature of the plant, that it is going to substantially damage your neighbors' plant. You can use the herbicide and some disposable sponge paintbrushes to attack the problem in two ways. This plant roots where it touches, and the roots go very deep. Killing the leaves will not kill the roots, and thus not kill the plant. First step, find where roots emerge from the ground, and cut them very close to the soil. Within 5 minutes, paint that cut stub with the undiluted herbicide. You do this quickly before the stub starts to heal over to protect itself, in the hope the poison will be taken up and transmitted to the roots. Just to make it easier to get at those roots, pull away all the vegetation you can and dispose of where it can't root itself or seed again. When you have gotten to all the roots you can on your side of the property, using a larger sponge brush, do a broad painting of all the leaves you can get at. This probably won't kill the roots on your neighbor's side, but the poison will spread through the stems to some extent. 

Of course, this will have to be periodically repeated, at least on your side, and even if the neighbors take out the plants on their side, it will continue to re-emerge from roots you have not been successful in killing. Patience and perseverence is your motto. 

Moral: The best way to get rid of invasive plants is to never plant them.