My question is, do I need a commercial license for the framework even though I don't want to sell the software and the source code is free for all employees of the corporation?
As long as you can comply with the (L)GPL requirements of the Qt libraries, then you do not need to purchase a "commercial license" from the Qt Company. Note that the terminology "commercial license" is a bit ambiguous, because the (L)GPL license also allows 'commercial' use as long as you can follow the obligations. Namely, the obligation for the LGPL is that you distribute the source code of the LGPL libraries themselves (i.e. the Qt libraries), and also that you allow your users to replace the LGPL libraries with alternative versions.
Basically what this has been interpreted to mean is that if you distribute LGPL libraries (such as Qt) with your application, then you also must:
Distribute to your users the source code of the same version of that library, or make it available in a way that is conformant with the LGPL license.
Make it possible for a user to use a different version of that LGPL library with your application. In practice this means that you must use dynamically linked libraries for the LGPL library (i.e. on Windows, the Qt libraries must be DLLs, and on Linux the libraries for Qt must be .so's).
In practice it also means that on platforms like iOS and Android, LGPL is not really usable (because these platforms apparently do not allow users to replace the libraries with another version). Basically, the LGPL requires that you document the process for replacing your version of the LGPL library (Qt) with another version that the user may have built herself.
Note that some components of the Qt framework may be GPL licensed, and not LGPL. If you don't want to release the source code of your whole application, you must avoid distributing GPL applications in a way that combines it with your application. To be safe it would be best not to distribute the GPL components at all together with your application.
See also: Can I use an LGPL-licenced library in my commercial app?