I have access to a company\'s unique linked in ID and would like to be able to do a search for its job postings; however, the only company-related input for the job-search
You can do this a couple of different ways. In neither case do you need to
Using job-search as you noted, you can do this:
http://api.linkedin.com/v1/job-search:(jobs:(id,company:(id,name),position:(title,location,job-functions))&company-name=LinkedIn
Note that job-search is a fuzzy match in this case so if you're close, it's probably good enough. You don't want to include "Inc" if you're looking for a company, though. The documentation for the job-search API is here: https://developer.linkedin.com/documents/job-search-api
Check the Hacker Summary at the top of the page to see all the different fields you can retrieve with a single call (there's no need to make subsequent calls to get more details).
However, if you've gotten the company ID you can get jobs directly from the job lookup API: https://developer.linkedin.com/documents/job-lookup-api-and-fields
http://api.linkedin.com/v1/jobs/1337:(id,company:(id,name),position:(title,location,job-functions)
Again, the same field selectors apply here. You can request the details for the job in the same query you use to find the jobs in the first place.