Yes, it is probably illegal. Yes, it is likely the neighbors will get it shut down. They are running a RV park, which has many licensing requirements. In Montana, for example, there are fire codes, plumbing codes, electrical codes sanitation codes and water testing that parks must meet and are periodically inspected for.
To build a park, you must apply, pay for, and get approved "subdivision for rent" for each site. Part of that process are environmental studies, impact studies and public hearings. The land must be properly zoned and taxed for business use. Making rural land an RV park will likely mean tax exemptions for agriculture, forestry, undeveloped rural land and the like will no longer apply to the property and the owners would have to pay higher taxes, just like every other business. And that is the requirements in one of the most rural states in the US.
Can you really blame the neighbors? They likely bought rural property for privacy and peace and quiet. A RV park next door isn't what they bargained for. The density of a park means much more traffic, much more noise, an increase in trash and generally they are an eyesore compared to forested land and farmers fields. The problems are further compounded by the fact the owners apparently don't actively manage the park and some bad tenants impacted those neighbors negatively.
As for increasing your cell service, yes there are products that will improve marginal cell service. Wilson is a popular brand, their website is wilsonamplifiers.com. Be aware that no matter what you use, you aren't going to have as good a connection you would have in an area with solid cellular service to begin with.
