You won't find this kind of count directly into a VCF file. But there is the allelic depth for the reference and the alternative alleles.
For example for this SNP:
chr_1 21682 . T C 150.0 . AC=1;AF=1.00;AN=1;DP=4;FS=0.000;MLEAC=1;MLEAF=1.00;MQ=56.44;QD=31.06;SOR=3.258 GT:AD:DP:GQ:PL 1:0,4:4:99:180,0
The reference allele is T and the alternate allele is C. And if you look at the AD (allelic depth) field, you will find that there is 0 reads supporting the reference, and 4 reads supporting the alternate allele.
You can use a tool like SnpSift extractFields to get this field.
Adding to Guillaume's response, you can also use samtools mpileup to get the full set of alleles from all the reads covering a genomic position (http://samtools.sourceforge.net/mpileup.shtml).
I have written a python script that converts the mpileup output to a table containing read counts.
Could you provide some more information? For example what program did you use and which parameters?
Thanks Guillaume an Noushin for detailed responses. Now, I think I can find what I need.