It is. A VCF ought to be sub-settable to a specific list of samples. The only thing you have to come up with for your specific case, is a list of 1K random sample IDs within the original VCF.
Geenerally speaking, knowing that you can do this and how involves getting familiar with the VCF specification as well as the main bioinfo tool used to manipulate and work with this format: BCFtools
Let's use 1000 genomes data as an example.
You can use a bit of UNIX together with BCFtools in order to get a list of 1K random IDs from individuals:
(sort -R instead of shuf might work too)
bcftools query -l ALL.chr22.phase3_shapeit2_mvncall_integrated_v5a.20130502.genotypes.vcf.gz | shuf | head -1000 > myRandomIDs.txt
Then again BCFtools to subset the list of individuals from the original VCF and into a new VCF file:
bcftools view --samples-file myRandomIDs.txt ALL.chr22.phase3_shapeit2_mvncall_integrated_v5a.20130502.genotypes.vcf.gz -o myNewVCF.vcf
If you dig a bit deeper into the BCFtools documentation, you will find many things you can do extra if you wanted, like re-calculating the allele frequencies based on this new cohort of individuals, if you wanted that too. Or subset for positions/variants, instead of individuals.
Using GnomAD data, selecting 1K random samples is impossible since data is aggregated and instead of genotypes per individual (and individuals you can select thus) we only have the frequency of the variants across various sub-cohorts within the GnomAD project. No individual level data.
The most you can do is select one of such sub-cohorts and filter for variants with a frequency > 0 (or a range) in such cohort. You could get variants present only and only is a specific cohort too, by using either frequency information or other info such as allele counts, etc.
The example below filters for variants with a frequency above 0 in the South Asian cohort and 0 frequency in the Southern European cohort. You can have a look at the header of the VCF in order to get to know all cohorts included and the meaning of all the information contained within the INFO filed in the VCF
bcftools filter -i "AF_sas > 0 & AF_nfe_seu = 0" gnomad.exomes.r2.1.1.sites.21.vcf.bgz
But I'm afraid that's not exactly what you wanted to do in the first place.