To determine how many t-shirts Emma needs to purchase, we first need to find out how many students are members of the spirit club.
Given that there are 300 students in the school and 20% of them are members of the spirit club, we calculate:
\[ \text{Number of spirit club members} = 300 \times 0.20 = 60 \]
Thus, Emma needs to purchase 60 t-shirts for the spirit club members. The options you provided (20, 60, 120, 300) indicate that the correct answer is 60.