First JavaFX is not using the LGPL but partly under:
The net effects is that JavaFX as a whole is not open source. Instead this is using a proprietary "trialware" license (the Oracle BCL) with some UI parts under the GPL + classpath.
Per the FAQ:
At this time, the JavaFX UI Controls source code has been contributed to the OpenJFX open source project; other JavaFX components are expected to follow in multiple phases. The code is available under the GPL v2 with Classpath Exception license, similar to other projects in OpenJDK. The Oracle JavaFX runtime and SDK implementations will continue to be released under the Oracle Binary Code License for the Java SE Platform.
So what can you do?
- if you can build everything in OpenJFX from sources and everything has been released by Oracle, you could build this and use these builds. I have not enough understanding of the JavaFX code nor what has been really released under OpenJFX to draw any conclusion. If everything is available under the GPL+classpath, you can read this answer to understand what would be your obligations under this license.
- otherwise, you cannot use much of JavaFX beyond the limited rights granted by the Oracle BCL, and you likely will need to buy a commercial license from Oracle.
Additional answer elements:
It looks like after all there is a GPL+classpath version of the JavaFX runtime that has landed in mercurial.
So what you could do is redistribute an OpenJDK and build yourself this JavaFX from sources as well as the libraries. If you rely on the pre-built binaries provided by Oracle as part of the BCL-licensed Java runtime, you are subject to the vagueness of the BCL and the fact this is not licensed for "function specific" usage.
The use of Software in systems and solutions that provide dedicated functionality (other than as mentioned above) or designed for use in embedded or function-specific software applications, for example but not limited to: ....
So if you opt for the BCL you will likely subject yourself to this kind of audit as Oracle can consider as function specific whatever it wants... And the word on the street and that it surely uses and abuses of this.
See this answer and this answer and this answer related to the BCL and Java and why it cannot be used for redistribution with free and open source software as well as commercially licensed software,
unless you either get a truly open OpenJDK (eventually building it yourself) or you buy a commercial license from Oracle.