Use a while loop that iterates through all files found by find:

#!/bin/bash

## counter starting at 1
COUNTER=1

find . -name "snps.vcf" \
| while read p
  do
  Basename=$(basename ${p%.vcf})
  mv $p ./targetdirectory/${Basename}${COUNTER}.vcf
  COUNTER=$((COUNTER + 1))
  done

This will scan for snps.vcf in the directories and then move them to targetdirectory appending an increasing number like snps1.vcf... until all files have been moved. Try to understand the code and ask if there are things unclear. If you are not familiar with | this is a pipe symbol, very important basic operator, google it if unknown.