Bouldering shoes are the most important piece of gear you will buy as a new climber, but you do not need to spend $150+ to get something that performs well. The shoes below all cost under $100, fit most foot shapes, and have the basic features that matter most: sticky rubber, a comfortable last, and a snug heel.
What makes a shoe "budget"? Mostly: simpler rubber compounds, less aggressive lasts, and lower-spec uppers. None of those things matter much for V0-V4 climbing, which is where every new boulderer spends their first year.