Over the last ten years high-throughput sequencing has enabled increasingly quantitative measurements of the diversity of lymphocyte receptor repertoires. A striking finding of these sequencing efforts has been that the clone sizes of cells sharing the same receptor are heavy-tail distributed. Here, we present a simple neutral birth-death model for immune repertoire formation in which all cells compete for a global resource. Homeostatic control of proliferation leads to a founder effect, in which large clones emerge early when there is less competition. We show that this mechanism produces a transient but long-lived regime of power-law scaling of clone sizes. A reanalysis of a cohort study shows that indeed early founded T cell clones are over-represented among the most abundant clones. We use data about how the founder effect diminishes over time to constrain how much peripheral selection impacts immune repertoire dynamics. Overall, our work suggests that dynamical processes early in life have a strong and long-lasting influence on the structure of the immune repertoire.