"People with only one abnormal gene are called carriers (heterozygotes) and may have mild, but medically insignificant, abnormalities of copper metabolism." 
However, with the heterozygous form of some diseases these "insignificant" abnormalities can be significantly exacerbated by an environmental trigger. For an example of what I mean, lets look at Factor V Leiden deficiency. If you have 2 copies of this gene your risk of venous thrombosis is 80%, but if you have one copy your risk is only 5-7%. HOWEVER, when you add an environmental trigger, in this example it's oral contraceptives, the risk of venous thrombosis jumps up to 30-35% for people with heterozygous Factor V Lieden deficiency. The normal population's risk when taking oral contraceptives is only 4%.
So clearly, this medicine is not safe for people with the heterozygous form of this disease. And interestingly enough, this is also an example of a genetic mutation that occurs in men and women equally, but has a gender-specific trigger (since men don't take OCPs).
My daughter has heterozygous Factor V, but the ONLY reason we know that is because one of her aunts had problems with blood clots, which eventually led to the Factor V diagnosis and a recommendation that all family members should be tested. If not for the advice to have genetic testing done, we may have found that my daughter has this condition the hard way; after she developed a blood clot due to OCPs.
The point I'm trying to make is that heterozygous Wilson's Disease doesn't seem to cause problems in general. However, studies have shown that children who are heterozygous carriers of Wilson's Disease do have problems with metal-homeostasis, and therefore it is quite reasonable to wonder if receiving mulitple doses of vaccines that contain toxic metals like aluminum and mercury would cause health problems in this genetically subsceptible group.
"It should be noted that individuals with genetic disorders affecting copper metabolism (e.g., Wilson's disease, Indian childhood cirrhosis, and idiopathic copper toxicosis) may be at risk for adverse effects of chronic copper toxicity at significantly lower intake levels [than the tolerable level of intake for the general population]."