We've been trying to contact Wolfram to purchase Mathematica license for our organization however we could not get any response. Does anybody have an idea? Is there anything wrong with the company or do we misunderstand something about the procedure? (I don't know, maybe we should first download the trial version and then purchase the license from there or something else?)
WR employees do not always see messages here. You should call again, and insure you have the current/right phone.
You can order normal licenses online "immediately". If you wish a "hard copy" perhaps order from a (book selling) store (unsure if these are still sold).
There's nothing wrong with a trial. Note if you buy then have to install "a new/full versions" to follow instructions for uninstalling before installing new (just 2 steps).