I don't know about the Doha Bank card, but I have the Masraf Al-Rayan card. (I assume we're talking about one of those prepaid cards.)

You apply for it at one their branches (I went to the City Center branch); you can print the form off and fill it in advance if you like, or do it at the branch itself. Google for: Pay and Go Prepaid Cards Masraf Al Rayan

After applying for it I was told I would receive a notification via SMS. Once I got that (after a few days) I was able to pick it up at the branch.