I realize you don't want to run random unsigned code from random places on the internet, but this seems pretty reputable - it's the ROES (internet upload utility) for ProDPI, who are a pretty well-known photo lab.

When I try to run it, I get "Fatal: Initialization Error: Could not initialize application"

For details, I get:

net.sourceforge.jnlp.LaunchException: Fatal: Initialization Error: Could not initialize application. The application has not been initialized, for more information execute javaws from the command line.
at net.sourceforge.jnlp.Launcher.createApplication(Launcher.java:813)
at net.sourceforge.jnlp.Launcher.launchApplication(Launcher.java:532)
at net.sourceforge.jnlp.Launcher$TgThread.run(Launcher.java:936)
Caused by: net.sourceforge.jnlp.LaunchException: Fatal: Application Error: Cannot grant permissions to unsigned jars. Application requested security permissions, but jars are not signed.
at net.sourceforge.jnlp.runtime.JNLPClassLoader$SecurityDelegateImpl.getClassLoaderSecurity(JNLPClassLo ader.java:2393)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.setSecurity(JNLPClassLoader.java:321)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.initializeResources(JNLPClassLoader.java:734)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.<init>(JNLPClassLoader.java:285)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.createInstance(JNLPClassLoader.java:357)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.getInstance(JNLPClassLoader.java:429)
at net.sourceforge.jnlp.runtime.JNLPClassLoader.getInstance(JNLPClassLoader.java:403)
at net.sourceforge.jnlp.Launcher.createApplication(Launcher.java:805)
... 2 more

Of all that, I'm guessing the salient part is:

Caused by: net.sourceforge.jnlp.LaunchException: Fatal: Application Error: Cannot grant permissions to unsigned jars. Application requested security permissions, but jars are not signed.

I've seen ways to turn that protection off, but is there a way to fix the situation without doing that? I updated everything with apt-get update yesterday, but it did not help.

Thanks in advance!

So you a did a web search for that error and found something like this.
If you are not prepared to downgrade your java security, then you need the other end to upgrade their security.

That's exactly the page I found.

Glad to know that I wasn't missing something glaringly obvious =)

So I made the recommended change, but it still fails. Now it gives me this:

"Fatal: Launch Error: Could not launch JNLP file."

Details:
net.sourceforge.jnlp.LaunchException: Fatal: Launch Error: Could not launch JNLP file. The application has not been initialized, for more information execute javaws/browser from the command line and send a bug report.
at net.sourceforge.jnlp.Launcher.launchApplication(Launcher.java:580)
at net.sourceforge.jnlp.Launcher$TgThread.run(Launcher.java:936)
Caused by: java.lang.reflect.InvocationTargetException
at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:62)
at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
at java.lang.reflect.Method.invoke(Method.java:498)
at net.sourceforge.jnlp.Launcher.launchApplication(Launcher.java:574)
... 1 more
Caused by: java.lang.NoClassDefFoundError: javafx/application/Platform
at com.softworks.roes.app.ROESApp.<init>(Unknown Source)
at com.softworks.roes.app.ROESApp.main(Unknown Source)
... 6 more
Caused by: java.lang.ClassNotFoundException: javafx.application.Platform
at net.sourceforge.jnlp.runtime.JNLPClassLoader.loadClass(JNLPClassLoader.java:1562)
... 8 more

Both your posts show the same error.

There appears to be some initialization step that has been skipped. Do a little more research on how to setup and start that app and follow all the steps. Once properly initialized it should start.

Even within the error message it gives you steps to follow.