I'm working as a researcher/bioinformatician looking for the genetic causes of rare diseases. Your chances of success are ... complicated. However, there are various things you can do.
I'll second the suggestion to use Exomiser, or its more expansive version called Genomiser.
No, it is not sufficient to look up the RS numbers in the VCF file. There are two reasons for this:
1. The RS number just refers to the location. Different variants can exist at one location, so you aren't necessarily finding the same variant. Variants need to be matched by location and by the change that they cause.
2. RS numbers are typically given to locations that have common variants, although there are numerous exceptions. It is a universal rule of genetics that a rare monogenic disease cannot be caused by a common variant. This fact was so obvious, but it needed to be published [0] before people started taking it seriously. So mostly likely the variant that is causing the disease does not have an RS number.
The main problem you will face is the sheer quantity of data that you have been given. The average person has something like 3 million variants, so you need a way to whittle these down to a short list. The first thing you need to do is get rid of all the common variants, for the reason stated above. The easiest way to do this is to annotate the variants using software like VEP, Annovar, or alamut-batch. I'd recommend VEP because it is good, popular, and free. That will include in its output whether the variant has been found in the GnomAD project [1], which is a conglomeration of thousands of genome sequences, and can therefore say whether the variant is common or rare. For the variant to be considered rare, it shouldn't be present in GnomAD more than a couple of times.
Once you have the variants annotated, you should know for each variant whether it is inside a gene, which gene that is, and whether the variant has an effect on coding. If a variant is intronic, it is unlikely to be pathogenic (although it is never that simple). Common mechanisms of pathogenicity are:
1. If the variant changes the protein code (a missense variant). These are hard to interpret - they may be pathogenic but most are not.
2. If the variant changes the length of the coding DNA by a factor of three (an in-frame indel), which inserts/deletes amino acids from the protein. These are slightly more likely to be pathogenic than missense variants, but most are still not.
3. If the variant changes the length of the coding DNA by something other than a factor of three (a frameshift indel). This messes up the frame of the three-base code of the gene, making the rest of the gene gibberish. These are much more likely to be pathogenic, but only if the gene itself is actually important.
4. If the variant changes a protein codon into a "stop" codon (a "stop gain" or "nonsense" variant). These are as likely to be pathogenic as a frameshift.
5. If the variant interferes with splicing (a splicing variant). These variants are on the borders of the exons of genes and may change the way that the introns are cut out of the gene before translation into protein. These are fairly likely to be pathogenic.
The annotations should tell you which of these things a variant might be. A synonymous or intronic variant that doesn't affect splicing is very unlikely to be relevant.
You need to determine whether the disease is likely to be recessive or dominant. Recessive means that you need to have both copies of the gene broken in order to get the disease, whereas dominant means that you need just one copy broken in order to get the disease. If you look the disease or gene in ClinVar or OMIM [2] you can often find whether the gene is recessive or dominant. If it is recessive, you either need to find two pathogenic variants that are heterozygous, or you need to find a single pathogenic variant that is homozygous. In the VCF file, a variant is heterozygous if it says "0/1" and homozygous if it says "1/1".
By far the easiest way to narrow down the extremely long list of variants is to do an inheritance analysis. If you are able to perform genome sequencing on both of the patient's parents, then you have more power. Namely, any variant that is heterozygous in one of the parents can't be causing a dominant condition in the patient if the parent is healthy. Any variant that is homozygous in one of the parents can't be causing a recessive condition in the patient if the parent is healthy. So, immediately reject any variant that is homozygous in one or both of the parents. Next, identify the variants that are only in the patient and not the parents. These are "de novo" variants - they arose in the patient as a copying error from the parent's DNA. A large proportion of rare genetic diseases are caused by de novo variants.
Other types of inheritance are:
1. Compound heterozygous - in this case one parent has one variant, and the other parent has the other variant, both in the same gene, and the patient has inherited both of them.
2. Homozygous - if both parents have the same heterozygous variant.
3. X-linked - if the patient is male, he has only one copy of the X chromosome. The mother may have a heterozygous variant on the X chromosome and be fine because of her second working copy of the gene, but pass the broken copy on to the patient. The father must not have this variant and be healthy.
There are more.
If you think you have found the causative variant(s), then you need to go through a process of proving it. The problem is that we have so many variants that if you look at the whole genome, you will find something, even in a healthy person. When we analyse someone in our lab with the parents available, we will typically produce a list of 20 genes that have some convincing arrangement of variants. The first hurdle that they need to pass is whether the gene is associated with the correct disease at all. If something is convincing, then a good guide to proving it is the ACMG guidelines [3]. These show how much evidence is required to classify a variant as pathogenic, and how to assemble that evidence.
Be very careful as a non-geneticist. Because we have so many variants, it is very easy to pick one and believe that it is the cause. The prior probability is that it isn't, unless you can gather significant evidence that it is. Early genetics studies tended to assume that if something was found, it must be the cause, and we are now paying for that. My lab recently published a paper refuting the association of some genes with a disease, because those associations were made back when standards were not as high and we did not have access to the population databases like GnomAD that we have now, and they were just wrong. If you think you have found the cause, then you will absolutely need to get it checked by someone qualified.
I wish you the very best of luck.
[0] https://www.nature.com/articles/gim201726
[1] https://gnomad.broadinstitute.org/
[2] https://www.omim.org/
[3] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4544753/