By my understanding, autogamy, i.e., reproduction that is sexual, but with both gametes contributed by the same individual, is not at all uncommon in the plant kingdom.
Does this also occur anywhere in the animal kingdom?
Yes, there are some animals which are known to self-fertilize (although generally not exclusively). For instance, among invertebrates some nematodes are known to use self fertilization, as discussed in this paper about its evolution. Among vertebrates, it seems to be very rare, apparently known only only in a single genus of fish.
It sounds like you're referring to self-pollination. This is a specific subset of the larger category autogamy, in which two gametes from the same individual fuse.
One source thinks that plants are the most successful example of autogamy because the development of several pollen tubes simultaneously allows the plant to carefully choose the fertilizing source, enabling it to select for genetic material that can mask the deleterious recessive alleles.