If I've understood correctly your question, I'd suggest you the following workflow:

1. Take the positions of those unannotated SNP.

2. Create a bed file with:

   SNP_chromosome    SNP_position-10000    SNP_position+10000
   

3. Create gene .bed|.gtf|.gff3 file with gene coordinates.
4. Intersect (bedtools intersect) these two files.

You can use the Ensembl VEP for this. To get variants 10kb up and downstream you'll need to alter the upstream/downstream distance to 10kb, which you'll find in the options.