write away for
1. To send a typed or handwritten request by mail. Typically followed by "to someone or something." We'll have to write away for the component we need, as there's only one manufacturer in the region and he doesn't have a phone or email address. Once I had enough barcodes from the back of the cereal box, I had to write away to the company in order to claim my prize.
2. To write a large amount very quickly or continuously. I want to get my thesis done soon, so I've been writing away for the last three weeks. I just want to put my headphones on and write away by myself—I can't abide all these distractions!
write away for something
to send for something in writing, from a distant place. I wrote away for a book on the rivers of the world. You will have to write away for another copy of the instruction manual.