I think I have solved this problem, with help from my apache error log (/var/log/apache2/error.log).
The log indicated that I was missing some directories where the plugin was attempting to move uploaded files to. To fix it I manually created two directories:
wp-content/uploads/woo-price-calculator/docs AND wp-content/uploads/woo-price-calculator/style.
After that the error indicated that I was missing two php modules. To fix this I manually installed (sudo apt-get install):
php7.0-zip AND php7.0-xml
After installing these modules and a restart of Apache the calculator upload seems to be working.