Juno will fly by Earth on October 9, 2013. There is an event to encourage Amateur Radio Operators to say high to the satellite. What I'm trying to do is figure out if I have any chance of pulling it off. In order to do that, I need to know the distance to Juno at a particular moment in time. How might I best be able to figure this out?
The JPL HORIZONS website is what you want for precise, tabulated values. Using the web interface, you can specify your position, and get all the info you could need for a given body (in your case, observer-centric (?) range to JUNO). It will also give you relevant parameters if you wanted to try and (visually) observe it, such as pointing angles, and the Sun-Spacecraft-Observer angle.
This site also writes to say
Hope this helps! I'm QRU unfortunately )+: