As previously noted, you don’t have to use the online app to send your document to your Kindle — on the right side of the page are various buttons that, when clicked, will offer instructions on how to send documents to your Kindle account via a mobile app, email, or various other methods.
Sending via email offers an extra added attraction: if you don’t want to struggle with trying to read a PDF on your phone (which, depending on the file, can be a real headache), you can use email to convert the PDF to a more readable format on your Kindle app. We’ve got instructions on how to do that here.