There are several components to this question:
Is hydrogen peroxide (as used in dental hygiene procedures) absorbed into the body in significant quantities?
Could absorbed hydrogen peroxide have systemic effects?
To what extent is oxidative stress a factor in the pathogenesis of vitiligo?
Can absorbed hydrogen peroxide contribute to the pathogenesis of vitiligo due to an increase in oxidative stress?
I will provide some evidence and discussion for each of these points below.
Hydrogen peroxide is the simplest peroxide (a compound with an oxygen–oxygen single bond). It is used as an oxidizer, bleaching agent, and antiseptic. Concentrated hydrogen peroxide is a reactive oxygen species and has been used as a propellant in rocketry. Its chemistry is dominated by the nature of its unstable peroxide bond.

1: Use of H2O2 in dentistry
This paper is a review of the use of hydrogen peroxide in dentistry.
The levels of H2O2 in toothpastes and dentifrices are generally much lower than those used for teeth whitening or bleaching.
Studies in which 3% H2O2 or less were used daily for up to 6 years
showed occasional transitory irritant effects only in a small number
of subjects with preexisting ulceration, or when high levels of salt
solutions were concurrently administered.
In contrast, bleaching
agents that employ or generate high levels of H2O2 or organic
peroxides can produce localized oral toxicity following sustained
exposure if mishandled.
They note potential health concerns raised by animal studies.
From a single study using the hamster cheek pouch model, 30% H2O2 was
referred to as a cocarcinogen in the oral mucosa. This (and later)
studies have shown that at 3% or less, no cocarcinogenic activity or
adverse effects were observed in the hamster cheek pouch following
lengthy exposure to H2O2.
So concentrations below 3% seem unlikely to cause significant harm. Beneficial effects (such as improvement in gingival disease and promoting healing due to an antimicrobial effect) are seen at concentrations above 1%.
Based on this, a window of 1-3% concentration of H2O2 would appear to be optimal.
A UK government toxicological report confirms that H2O2 only causes local toxicity by irritation and cellular damage, but it does not cause systemic problems and is not considered a carcinogen.
2: Metabolism of H2O2
H2O2 is a byproduct of many biochemical reactions and is toxic to cells. The enzyme catalase is found in nearly all living organisms exposed to oxygen (such as bacteria, plants, and animals). It catalyzes the decomposition of hydrogen peroxide to water and oxygen. It is a very important enzyme in protecting the cell from oxidative damage by reactive oxygen species (ROS). These are also known as free radicals.
The reaction is shown below.

The toxicity of H2O2 is due to oxidation of proteins, membrane lipids
and DNA by the peroxide ions. The class of biological enzymes called
superoxide dismutase (SOD) is developed in nearly all living cells as
an important antioxidant agent. They promote the disproportionation of
superoxide into oxygen and hydrogen peroxide, which is then rapidly
decomposed by catalase to oxygen and water.
What is oxidation?
Oxidation is one of a pair of opposite chemical reactions involving transfer of electrons; oxidation and reduction. Together they are known as redox reactions.
Oxidation is the loss of electrons or an increase in the oxidation state of an atom by another atom, an ion, or a molecule.
Reduction is the gain of electrons or a decrease in the oxidation state of an atom by another atom, an ion, or a molecule.
In a redox reaction, one component is oxidised and the other reduced.
For example, during the combustion of wood, electrons are transferred
from carbon atoms in the wood to oxygen atoms in the air. The oxygen
atoms undergo reduction, gaining electrons, while the carbon atoms
undergo oxidation, losing electrons. Thus oxygen is the oxidizing
agent and carbon is the reducing agent in this reaction.
Another example of a redox reaction is the oxidation of glucose to release energy in cellular respiration.
Catalase uses the oxidising power of H2O2 to oxidise other potentially harmful substrates that need to be broken down (like phenols, formic acid, formaldehyde, and alcohol), while the H2O2 itself is reduced in the process. This is called the peroxidation reaction.
Catalase is present in all cells (especially the liver and kidneys). Apart from local tissue destruction, small amounts that might be absorbed would be broken down in this manner.
Vitiligo is a condition in which melanocytes in patches of skin cease production of the pigment melanin.
There are several theories about its pathogenesis. It is associated with certain genetic susceptibility markers and also with other autoimmune disorders, suggesting an autoimmune process. The autoantigen is likely to be the tyrosinase enzyme, which is involved in melanin biosynthesis.
Oxidative stress theory
This research paper discusses oxidative stress in the context of vitiligo.
In vitiligo, melanocytes are particularly vulnerable to oxidative
stress owing to the pro-oxidant state generated during melanin
synthesis and to the genetic antioxidant defects.
In particular, with relevance to catalase:
In both lesional and non-lesional skin of vitiligo patients, increased
levels of reactive oxygen species (ROS) were observed, accompanied
with reduced levels and activity of catalase.
It is well known that normal melanocytes are vulnerable to excessive
reactive oxygen species (ROS) for their specialized function of melanin synthesis, which would
result in a highly pro-oxidant state in the epidermis.
This imbalance between the pro-oxidants and the antioxidant status can
disrupt the homeostasis of melanocytes, causing accumulation of
oxidized and damaged proteins or organelles, and consequently lead to
the destruction of melanocytes.
So there is evidence of increased oxidative stress in the pathogenesis of vitiligo. However, it is a complex picture, with probable environmental triggers and genetic susceptibility factors. An autoimmune process and oxidative stress are not mutually exclusive, as the former could disable the cellular mechanisms for dealing with the latter.
4: Can absorbed hydrogen peroxide contribute to the pathogenesis of vitiligo due to an increase in oxidative stress?
Essentially, assimilating this information, the answer seems to be no.
Applied hydrogen peroxide (H2O2) primarily causes local irritation and tissue damage, depending on the concentration. It is not significantly absorbed and small absorbed amounts would be quickly broken down by catalase.
While increased susceptibility to oxidative stress in melanocytes plays a role in the pathogenesis of vitiligo, it is a disruption of the cellular mechanisms for managing oxidative stress, rather than a primary increase of oxidants (like hydrogen peroxide) that is the main factor.