This, I believe is a well known concept within Nichren Buddhism (and perhaps other parts of Buddhism; I don't know), but I am wondering if there is a name for the idea that the environment and self are a single entity. This is to say that I am in the same entity as my environment and so is everyone else. This is also to say that one's environment is a reflection of one's self, not the other way around and that in order to change one's life, one has to change one's self. Is there a name for this philosophical movement or anything that resembles this?
Not the movement, but the transitory perception of that, in modern Western buddhism/Eastern-spiritual-practices followers and the psychedelic community is referred to as "ego death". I don't think there is a movement solely for that, it is more of a paradigm found in both ancient Eastern philosophies/religions and, with the advent of Ram Dass, a few modern 'schools of thought'.
The way the question was asked, Virmaior's answer of monism or holism in the comments above seems correct to me. However, the Unitedness of environment and self, or more commonly, oneness of person and environment, is defined quite differently than "I am in the same entity as my environment and so is everyone else." Below there is an excerpt from, and a link to, a Nichiren Buddhist dictionary entry for oneness of person and environment to compare. (I do like the term "unitedness" by the way.)
Note that an alternate translation of "oneness" is "nonduality." Consequently, there are few similar ideas in modern philosophy since mainly monisms or pluralisms are considered. The very fact that nonduality is such an awkward word in English is a big clue that English speakers probably have not considered "Unitedness of environment and self" or other nondualisms. However, a relatively recent example which may fit your interest is Whitehead's Process Philosophy which some compare to buddhist thinking. Whether Whitehead has a term that covers oneness of person and environment I do not know.
oneness of life and its environment ［依正不二］ ( eshō-funi): Also, non-duality of life and its environment. The principle that life and its environment, though two seemingly distinct phenomena, are essentially non-dual; they are two integral phases of a single reality. In the Japanese term eshō-funi, eshō is a compound of shōhō, meaning life or a living being, and ehō, its environment. Funi, meaning “not two,” indicates oneness or non-duality. It is short for nini-funi, which means “two (in phenomena) but not two (in essence).” Hō of shōhō and ehō means reward or effect. It indicates that “life” constitutes a subjective self that experiences the effects of its past actions, and “its environment” is an objective realm in which individuals’ karmic rewards find expression. (excerpted from Oneness of life and its environment)
Western philosophy contains at least a couple flavors of this basic idea, as one might expect, given the people involved, these take on mechanical metaphors.
The most literal similar position that comes to mind is monism, and reflected by something like Leibniz 'Monadology' or Whiteheads "Process and Reality".
The model Leibniz expressed was that every element of the world is a reflection of all the relationships it has to everything else. His image is that a soul is like a reflective sphere, which captures reflects the image of all the spheres around it, but other than that has no interior or content. It is the set of reflections and nothing else.
Another closely related approach is the perspective of psychoanalysis, as captured by Jung, or by a group-dynamics perspective like that of Bion or the Tavistock movement.
The basic model of thinking here is that whatever affects come to you are ultimately recorded either in your conscious or in your unconscious mind. The things that are unconscious are still part of you, and will affect your behavior without your conscious consent. But the effects that pass between people also happen on conscious and unconscious levels. So in some sense, the collection of emotional and other effects people have on one another are a shared pool, and not a set of segregated bundles belonging to individuals.
Every group has a collection of shared material that is passed between them but for which no one in the group takes responsibility. This is the group's unconscious. Ultimately the parts of everyone's thoughts for which no one takes responsibility are part of a vast repository known as the Collective Unconscious.
This perspective bears up the 'if you want to change your surroundings, you change yourself' morality. By deciding what you consciously attend to and resolve, and what you let have its own way with things, individuals channel the pool of thoughts passing between people in one way or another, and that controls their entire social environment which controls the vast majority of their physical environment.