Bilinear map that is continuous on both variables is continuous
You have $|B_y(x) | \leq M_y \|x\|$. Think of $(B_y)_{y \in Y, \|y\|\leq 1}$ as a family of continuous linear maps on $X$. At each point $x\in X$ We have $\sup_y |B_y(x)| <\infty$. By Uniform Boundedness Principle this implies that $\sup \{|B(x,y)|:\|x\| \leq 1 , \|y\|\leq 1\} <\infty$. This implies continuity of $B(.,.)$.