The sentence you have quoted is a disorganized and half-broken sentence that only happens when someone says something as they are thinking. I feel particularly uncomfortable with this sentence because it has no copula (だ/です) corresponding to (通い始めた)理由が. An educated person never constructs a sentence like this when writing carefully.
However, in speech, people often have to start a sentence before deciding how to finish it. You have to get used to sentences like this. I think this sentence can be explained in several ways:
In casual speech, people often prefer て/で to です even at the end of a sentence (言いさし表現) to keep the listener's attention. When parsing a long sentence connected by many て, such as this one, it may be necessary to consider that a sentence has effectively ended after a て/で. See: て form at end of phrase but not being used for requests
As you said, sometimes people even start another sentence in the middle of one sentence. It's called a parenthetical aside (See: What does 人の手により in this sentence mean?). In your case, it seems that the speaker suddenly wanted to start to describe a reason by saying 通い始めた理由が, but then failed to correctly conclude the parenthetical part by saying something like なんですけど.
Even in serious writing, some people are not very good at matching a subject and a predicate correctly. See: Sentence topic without grammatical function
As for the implied subjects, the subject of キャンペーンをやっていて is either the school itself or someone in the school. Something like this is very common, and this is not a problem of this sentence.
I made minimal changes to the original sentence. The following is still a little messy as written language, but is organized enough as conversational sentences. For example, an editor of an interview article would revise the original transcription to something like this. If you feel comfortable reading this one, don't worry too much about the remaining "broken" parts:
10歳、11歳ぐらいのときに、子供向けの英会話教室に通い始めまして。通い始めた理由が、その英会話教室に入会したら、ぬいぐるみがもらえるっていうキャンペーンをやっていた、というものなんですけど、本当にそのぬいぐるみが欲しくて、母に『英語頑張るから』と言って、入会させてもらいました。