Can a brand do affiliate marketing without affiliate marketing software?

Technically, yes. For affiliate marketing, all you need is an audience and a bunch of partners who will advertise to your audience. 

Do you have a thriving YouTube channel, Instagram following, or Facebook community? You could get away with doing affiliate marketing on a small scale without a platform or software. However, you will need affiliate marketing software to tap into a larger audience, find affiliates, manage their performance, and track your returns from affiliate campaigns.