Is it possible to add an assignment to a course using a web service?
There are no function for that on the current list https://docs.moodle.org/dev/Web_service_API_functions but there are some discussions on tracker. I could not find the MDL issue, but I remember playing with that on the past.
I created a tracker for this, so please vote if you wish to have this or just want to help me
you may take a look at MDL-40779. I linked this to your issue.
I would recommend not mention that you "desperately" need this. A business or institution must understand https://docs.moodle.org/dev/Process and it's calendar. Your issue has more chances to be accepted if you can provide a code solution or initial work on code that will be reviewed by HQ. A business or person can always contact a Moodle Partner on his/her territory to get support/consulting/development services to help with any Moodle issues. There is also the Moodle User Association that serves as a channel to get things done on Moodle.
I removed the word 'desperately' from the tracker. We do understand how things work, yet sometimes there's a very long wait for even code fixes. We will probably get this build by a third party.
The tracker is posted to keep track on wanted functionality within Moodle. If the MUA picks it up, then we are happy, but we not always have the luxury of waiting for the fix.
I voted for MDL-40779 and will be watching it from now on.