Given files genes.bed and snps.bed, you could do something like:

$ bedmap --echo --echo-map-id --delim '\t' genes.bed snps.bed > answer.bed

The file answer.bed will contain the gene annotation and a semi-colon delimited list of SNP identifiers that overlap each gene.

In order to get genes.bed, you could use Gencode v44 (hg38):

$ wget -qO- https://ftp.ebi.ac.uk/pub/databases/gencode/Gencode_human/release_44/gencode.v44.annotation.gff3.gz \
    | gunzip --stdout - \
    | awk '$3 == "gene"' - \
    | convert2bed -i gff --attribute-key="gene_name" - \
    > genes.bed

To get snps.bed, you could use dbSNP (v151):

$ for chrom in `seq 1 22` X Y; do \
    wget -qO- https://ftp.ncbi.nlm.nih.gov/snp/organisms/human_9606_b151_GRCh38p7/BED/bed_chr_${chrom}.bed.gz \
        | gunzip ---stdout - \
        | sort-bed - \
        > hg38.snp151.chr${chrom}.bed; \
    done
$ bedops -u hg38.snp151.chr*.bed > snps.bed

Then run the bedmap statement above.

If you want SNPs that overlap regions only where genes overlap, then you can do the following, instead:

$ bedmap --echo --echo-map-id --delim '\t' <(bedops --intersect genes.bed) snps.bed > answer.bed

The file answer.bed will contain the intersection of regions where genes overlap and a semi-colon delimited list of SNP identifiers that overlap each intersection-region.

ref. https://bedops.readthedocs.io/en/latest/content/reference/set-operations/bedops.html#intersect-i-intersect