First introduced by Lascoux and Schützenberger in 1982, Grothendieck polynomials are a family of polynomials indexed by permutations in $S_n$. In this thesis, we study the supports of Grothendieck polynomials associated to vexillary (2143-avoiding) permutations. We use bumpless pipe dreams to show several nice properties of the supports of vexillary Grothendieck polynomials, addressing special cases of conjectures of Mészáros, Setiabrata, and St. Dizier. Additionally, through joint work with Mészáros, Setiabrata, and St. Dizier, we show that the homogenized Grothendieck polynomial associated to any vexillary permutation is M-convex. To accomplish this, we introduce bubbling diagrams and show that they compute the supports of vexillary Grothendieck polynomials.