The integrity of software builds is fundamental to the security of the software supply chain. While Thompson first raised the potential for attacks on build infrastructure in 1984, limited attention has been given to build integrity in the past 40 years, enabling recent attacks on SolarWinds, event-stream, and xz. The best-known defense against build system attacks is creating \emph{reproducible builds}; however, achieving them can be complex for both technical and social reasons and thus is often viewed as impractical to obtain. In this paper, we analyze reproducibility of builds in a novel context: reusable \emph{components} distributed as \emph{packages} in six popular software ecosystems (npm, Maven, PyPI, Go, RubyGems, and Cargo). Our quantitative study on a representative sample of 4000 packages in each ecosystem raises concerns: Rates of reproducible builds vary widely between ecosystems, with some ecosystems having all packages reproducible whereas others have \issues in nearly every package. However, upon deeper investigation, we identified that with relatively straightforward infrastructure configuration and patching of build tools, we can achieve very high rates of reproducible builds in all studied ecosystems. We conclude that if the ecosystems adopt our suggestions, the build process of published packages can be independently confirmed for nearly all packages without individual developer actions, and doing so will prevent significant future software supply chain attacks.