Short tandem repeats (STRs), genomic regions each consisting of a sequence of 1-6 base pairs repeated in succession, represent one of the largest sources of human genetic variation. However, many STR effects are not captured well by standard genome-wide association studies (GWAS) or downstream analyses that are mostly based on single nucleotide polymorphisms (SNPs). To study the involvement of STRs in complex traits, we imputed genotypes for 445,735 autosomal STRs into SNP data from 408,153 White British UK Biobank participants and tested for association with 44 blood and serum biomarker phenotypes. We used two fine-mapping methods, SuSiE and FINEMAP, to identify 118 high-confidence STR-trait associations predicted as causal variants under all fine-mapping settings tested. Using these results, we estimate that STRs drive 5.2-9.7% of GWAS signals for these traits. Our high confidence STR-trait associations implicate STRs in some of the strongest hits for multiple phenotypes, including a trinucleotide STR in APOB associated with LDL cholesterol and a CGG repeat in the promoter of CBL associated with multiple platelet traits. Replication analyses in additional population groups and orthogonal expression data further support the role of a subset of the candidate STRs we identify. Together, our study suggests that polymorphic tandem repeats make widespread contributions to complex traits, provides a set of stringently selected candidate causal STRs, and demonstrates the need to routinely consider a more complete view of human genetic variation in GWAS.