IGSR do not provide any tools for fetching data from a single gene. We recommend using Ensembl for fetching variants for a particular gene or locus. There you can search for a gene and find a variant table (for example) which you can filter to only show variants found in 1000 Genomes. You can also use the Ensembl Data Slicer to fetch a VCF or a BAM covering a particular genomic region.
You can also see the per-individual sequences of proteins and transcripts from 1000 Genomes in the haplotypes page (for example) for an Ensembl transcript, showing the variants that co-occur in a single transcript or protein, their sequences and which individuals they have been identified in.