Yes, you can stay in Kenai Fjords National Park, but lodging options are limited. There are no hotels or lodges within the park itself. However, you can camp at the Exit Glacier Campground, which is a walk-in tent campground with 12 sites available on a first-come, first-served basis. Exit Glacier Campground is open from May to September, weather permitting.
Additionally, there are numerous lodging and camping options in the nearby town of Seward, which serves as the gateway to Kenai Fjords National Park. You can find cabins, campgrounds, and RV parks in the area. Some options include fishing camps and forest campgrounds.