I don't think they can, Floral Park is an Incorporated Village and Elmont is an unincorporated area in the Town of Hempstead. So they'd have to bring suit against the Town of Hempstead and/or State of NY.
I think Floral Park may be concerned about the loss of parking revenue from Elmont Residents at the Floral Park station, I don't see how an Elmont Station affects them otherwise, maybe patronage to their business on Tulip Ave while going to and from the Train?
As for the Arena, they may be concerned with extra traffic in the area and those looking for on-street parking, Belmont Stakes is one day a year and that's enough to deal with in both Floral Park and Elmont's eyes, and even Queens Village, since they have to deal with it as well.
The new station will be on the main line, so trains will be able to enter and leave the station quickly, no need for speed restrictions like when going through the Belmont spur's wye, etc. They could easily serve the station during events by adding a stop to all the service they have on the main line. (both east and westbound).