It seems like an ideal situation. A job opens up at an organization you really want to work for, and it’s a good match for your skills. And– you actually know someone who works in the organization. Chances are, you probably don’t know the person that well. Maybe you worked with him or her a few years ago, or were classmates. You’re not even sure they remember you. Can they still help you get a job?
Typically, the timing on a job application is quick, while your outreach and networking can take time. It’s usually safer to just go and apply for a job, then follow up with your contact, just so you don’t miss an application deadline. (There’s one exception to this– if the organization has a hiring referral bonus program, your friend could earn cash for recommending you for the job. Most organizations in the nonprofit and government sectors don’t do this, but some for-profit companies do, and you might ask first before applying, then follow up for an informational interview.)
I think it’s fair to reach out to your contact via LinkedIn or email with a message briefly reminder her how you know each other, then noting that you’re very interested in working for her organization and would be interested to see if she might have a few minutes to chat with you in the next week or two about the position you’ve recently applied for. Then show your appreciation for her time. Hopefully she’ll get back to you. If so, then you should have a list of questions ready, to get a sense of the organization’s culture, how you would fit in to it, how the job became available, what they are looking for in the ideal job etc. You can also express your interest in the job and say why you think you’d be a good fit, though most of the discussion should be more of an informational interview rather than a sales pitch/job interview. Ideally, she will then volunteer to mention to the hiring manager that she’s spoken with you, and this will give your resume a second look if not an interview. If all goes well, the informational interview could even turn into a job interview and you’ll get the job.