Most of Dortmund's capacity is taken up by season ticket holders and sales of tickets to away fans usually go via the respective team's supporters club.
Your chances of paying on the day are non-existent this season, since the club have been doing so well, tickets only go on sale to the public a few weeks before the game because members have the first option on buying those left to non-season ticket holders.
I would recommend viagogo.de who have never let me down, they will sell Dortmund tickets for games almost 1 year in advance, they also promise you the ticket or your money back. The downside is that these will cost up to 6 times the face value.
Alternatively, try ebay.de who also have been a great source of tickets over the years.
Touting/scalping is not illegal in Germany, so you'll find a lot of people selling tickets at the U-bahn station by the stadium on match day.