Java 6 has reached EOL in February this year and will no longer receive public updates (including security) unless you buy very expensive enterprise support.
That should be all the reason needed.
Besides, overwhelming evidence suggests that backwards compatibility for Java runtimes is excellent. Chances are that you just have to replace the Java 6 installations with Java 7 and all applications will just continue to work without any problems. Of course this is not guaranteed and extensive tests are recommended to confirm that there will indeed be no problems.