You can certainly download the app onto your device without actually using it. But I assume you really want to know if you can open .mobi files in the Kindle app without an Amazon account. And I'm afraid the answer to that is no.
The good news is that you can use other apps to read .mobi files. FBREADER is one such reader which works on both ios and android; you can find other android apps here http://www.androidauthority.com/best-ebook-ereader-apps-for-android-170696/ . They just can't read the .mobi digital rights management locked files.
Also, remember that .mobi is in many ways an inferior ebook format to .epub . If at all possible, I always prefer the .epub file to the .mobi file (and open them in Google Play Books, iBooks, Adobe Digital Editions, whatever). If you don't have access to the Kindle store, then you can probably find the file as an ebook in another store.