Your best bet is to go to the astronomy department at your college or university and talk to the department chairman. Ask him/her whether they, or anyone in the department might know of a place where such a program exists. I know this sounds like an obvious first step, but usually the astronomers at your own institution would know where such internships are offered in your geographic area. Sometimes, astronomers may have a few thousand dollars available through their own research grants to support a summer intern. If you have tried this and come up empty handed, your next step is to contact some likely places yourself. It is good that you are starting early!
You need to decide just how far you are willing to travel to get to a hosting institution. Places like the major research observatories at Kitt Peak in Tucson Arizona, or the National Radio Astronomy Observatory ( VLA, Greenbank etc) in Charlottesville Virginia, may have such programs in place, but I suspect that the people that get first crack at them come from the neighboring University of Arizona or the University of Virginia. You might consider going to my, HOTLIST and looking through the list of professional resources and other categories. Look for the homepages to the major national observatories and institutions and contact the person/persons that authored the web page. They may be able to give you more information about what other programs they have, and whether they support summer internships.
I have also just come across an announcement for a program called the Research Experience for Undergraduates. Contact Dr. James Wright at the National Science Foundation at 703) 306-1819 for more information. In 1993 the places that were a part of this program and which offered internships were: University of Alabama, Cornell, Haystack Observatory, Maria Mitchell Observatory, National Optical Astronomy Observatory, National Radio Astronomy Observatory, Northern Arizona University, Pennsylvania State University and the University of Wyoming. The list may change each year. Good luck!!