Is there a PDF reader app for the Kindle Fire with Dropbox syncing? Are there any apps for reading PDFs on the Kindle Fire that sync with Dropbox?  (Ideally I'm looking for an app comparable to the excellent GoodReader for iOS.)

Note: Dropbox does not have an app in the Amazon Appstore.
If you allow non-Appstore apps to be installed in your device, you can install Dropbox from their website.  

To do that, 


Click the gear icon in the top-right corner of the Kindle Fire home screen to activate the Settings window
Click More
Scroll down to Device
Scroll down to ‘Allow Installation of Applications from Unknown Sources’ and move the slider from ‘Off’ to ‘On’ and confirm the selection


Then head to Dropbox's site and download the apk from their site (NB: since this is not attached to a market, you will not be notified of updates):  http://www.dropbox.com/android

Once that's installed, you should be able to use any PDF app that's installed on the Fire. 
{Begin biased statement} I'm partial to my company's own qPDF Notes and we have some nifty UI enhancements coming along shortly but {end biased statement} there are a number of other apps that will read PDFs on the Fire and, assuming they've set up their Intents (that handy little window that pops up in one app when you want to Send or Open or Share something in another app) properly, you should be able to click on a PDF from within Dropbox and open in the app of your choice and click Send from the app of your choice and save it to Dropbox.