It is not forbidden, although, customarily, some people might consider it better to wait.
Hadith focus on mourning on the day of Ashura and refraining from worldly affairs on Ashura, or for the first 10 days of Muharram, rather than the entire month of Muharram. I would personally advise against buying a house on the actual day of Ashura, although again it is not forbidden.
Anyway, you have to live somewhere, and if you need somewhere to live now, or if your best option to buy the house now, you have to make the choice that is best for you. You are better able to serve Imam Husain if you are not homeless!