Yes there is a Dropbox app for the Nook tablets, no there isn't an app for the e-ink Nook. No e-ink eReaders can install apps.
By rooting your Nook ereader you could install apps anyway, you may want to check that out. It might be easier than you think -- there are some good guides.
http://blog.the-ebook-reader.com/201...-or-nook-glow/
http://nookdevs.com/Nook_Simple_Touch/Rooting