I don’t know what is the exactly form of this verb in this sentence.
I would stay with the person until the emergency service arrives/arrive/arrived?
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I would stay with the person until emergency services arrived.
Note that "emergency services" is plural. At least in the US.
Also note that your use of "would" is ambiguous here. It could be a hypothetical situation or it could be a situation that happened in reality, over and over, in the past. The wording is the same but the meaning is quite different. When asking a question like this you might want to clarify which usage you are referring to, though in this case the answer is the same either way.