The ultimate goal when you meet up with a networking contact is to get information that will lead you to a nonprofit job. That doesn't mean you should straight up ask for a job. Instead, ask for references.
The fact of the matter is that career networking isn't always going to lead you to people who have solid job references to help your job search. Instead of putting your contacts on the spot about a potential job, mention to them what kind of positions you are most suited for and the ask if you can use them as a reference when you apply for jobs.
There are a couple of good reasons for taking this approach. First of all, being used as a reference is a pretty high compliment. It says that you think enough of this person to use them as an accurate judge of your abilities. It will also make them feel better about themselves knowing they are able to help you out in your quest to get a good job.
All of this sounds like it's better for your contact than for you, but these good feelings will make your contacts more eager to help you. They will think of you as a friend and will be thinking about other ways they can help you with your search. This means they will likely be on the lookout for jobs that fit your needs.
It's important to keep in mind that this technique works best with people that you already know fairly well. If you are just getting to know someone you discovered through LinkedIn, you should wait for a few conversations until you ask to use them as a reference.