I don't think there is currently an API call for that. A private API shard is expected at some point in the indefinite future, but currently even the private API shards are all read-only.
Unless [violet] is willing to take the time to add that, your best option is to write a non-API script that uses the same issue-answering URL human players do. Note that non-API scripts have
different rules from API-using scripts, the most important one of which (for your application) is that the ratelimit is 10 per minute instead of 100 per minute.
Note that you can make things a little easier on both your script and the server by appending /template-overall=none to the URL, for example
http://www.nationstates.net/page=dilemmas/template-overall=none. I'm not sure where I learned that, the linked script rules seem to make no mention of it.
Or more topically,
http://www.nationstates.net/page=enact_dilemma/dilemma=X/choice-Y=1/template-overall=none (don't click on that!), where X is the number of the issue, and Y is the internal number of the option (usually one less than the displayed number, so 0 for the first option, 1 for the second option, etc., but can be different if some options don't appear due to option validities, and -1 for dismissal).