It's the neighbor's place to call the lawn company. Get the number and follow up directly. Send lawn company a letter with reference to Fla Stat 627.4137 insurance disclosures. If they have liability insurance, then a claim should be opened and they should send you their coverage info. If so, then you follow through with the claim. If they ignore you or if they don't have coverage, then you can consider the costs/hassles of suing vs. fixing the fence yourself.
You might also want to put the neighboring homeowner on notice and request their HO policy info. The owner's mailing address should be listed on the county property appraiser's website.
If you bring your own HO policy into this, you will have to eat the deductible amount, which may exceed the cost of repairs.
Do you know the consequences of your legal situation on your Financial & Estate Plan? Dennis Phillips is an attorney and financial planner based in South Florida. He is a member of the Florida bar, he holds the nation-wide Series 65 Investment Advisor license, and holds an insurance license in Florida and Virginia. Disclaimer: The response above is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, would significantly alter the above response.