One of the things the 1000 Genomes (led by Richard Durbin's group) is doing is building a "read server", which effectively allows you to make web queries for primers (or kmers or haplotypes) and it will tell you which sample has them. Right now it's not yet publicly available, but it will be soon.

Right now I think your best option is to check the VCFs more carefully and see if your variant is there - you don't need to download all the VCFs (which are enormous) - you can just get the file that lists all the sites, and see if there is anything resembling yours

so what you want is to end up with just a single fasta sequence from a bam file, and then try to find your primers in it? if that is the case, you "only" need to convert the bam file into a fasta sequence and then perform some simple pattern matching.

first, get the variants out of the bam file and index them:

samtools mpileup -uf hg19.fa in.bam \
| bcftools call -vm -Oz - > out.vcf.gz
tabix -p vcf out.vcf.gz

then, extract the reference section you're interested in, build the consensus applying the variants, join all the fasta sequence lines into a single one per each fasta entry (for later searching purposes), and compress it in case it's expected to be a big file:

samtools faidx hg19.fa chr22:29000000-29200000 \
| vcf-consensus out.vcf.gz \
| perl -pe '/^>/ ? print "\n" : chomp' \
| tail -n +2 \
| bgzip > out.fasta.gz

finally, count the number of lines that match your primers:

zgrep -cf primers.txt out.fasta.gz

I used regions chr22:29000000-29200000 for this example, but any other region/s can be selected. have in mind that if you don't specify any region/s, the samtools faidx command will output the entire genome.

You might just convert the reads near that area back to fastq (e.g. with samtools bam2fq)and assemble that. If 1000 genomes found that deletion, then it might be easier to just look at the sample genotypes in the area neighboring the deletion and modify the reference sequence to match (you'd then design primers against that).