As of 2015, you can make reservations directly through the Federal Parks Service at Recreation.gov. On this site, you can reserve camping spaces, picnic tables and other facilities.
First, put the name of the park you would like to visit into the search box. For example: Mount Rainier National Park. The predictive drop-down box helps you find what you're looking for. Select your intended park, and then browse what is offered. On the left side, you can filter for campgrounds, permits, tours and other items. The right side shows a map with the location of each item. Click an item that interests you to see the specific camp spaces or facilities available to reserve, and then enter date and length of stay to see which specific facilities are available for your trip. From there, you can see the price of your reservation. Next, select Book Now.
When you are prompted to log into the website or create a profile, enter your email address, create a password, enter some personal information, and proceed to checkout. Recreation.gov accepts credit and debit cards, and sends you a confirmation email once you complete the booking. Check out the Help & FAQ tab at the top of the page if you need further assistance.