I have been on the main on-site interview at 2 companies. I feel I would be invited in for the second and final on-site interview. Their feedback has also been that I was the first in line and although they seem interested in me, they have to go through the other applicants. What can I (Should I) say that could get the hiring managers to act quicker? I am immediately available.
What can I (Should I) say that could get them the hiring managers to act quicker?
You could tell them that you need an offer within a week, or you will withdraw your application. That will get you a very quick answer, but it may not be the answer you are looking for. I probably wouldn't suggest that approach for most candidates.
Perhaps the best you can do is indicate that you are actively interviewing elsewhere. If for some reason the hiring managers have already decided that you are the most qualified candidate they have seen to date, they may choose to short-circuit their plans.
More likely, they won't speed things up if "they have to go through the other applicants" as you have indicated. Sometimes these things just have to play themselves out.
You'll have to decide if you can live with the process as it is, or move on to other potential jobs.
How do I speed up the hiring process?
If you're already interviewing with other companies or have another offer, telling the company about it may speed up their process, but only if you're a strong candidate anyway.
What can I (Should I) say that could get the hiring managers to act quicker?
Unless you have another offer, there's probably very little you can do and even less that you should do. Trying to rush the process may make you look desperate or like you don't understand how the hiring process works and neither of those are going to improve your application.