Our fridge is broken and we are desperate for a new one. Can we purchase one in the 9 days? What if we order it tomorrow (Thursday) and it arrives in the 9 days?

ב"ה Shalom This is the classic example of something you are allowed to buy even during the 9 days and to start using it. Since you are buying it out of necessity and not because you decided you want for your pleasure a newer fridge. (שו"ע סי' תקנ"א מ"ד ס"ק יא, פסקי תשובות יא All the best