There is about a dozen spots right at the trailhead, that usually fill up by around 2am (at least, when I go in early September). However there are many more within a minute or two walking, so not an issue.
Are you going to acclimate at the campground both nights?
Make sure to hike up slowly, within the limits of your conditioning. There is a high correlation between fitness and altitude sickness (since highly fit people need less oxygen at a given pace), but only if you stay within your limits. And that means keep your heart rate in a low zone.