。
Tears suddenly flooded from her eyes.
Literal
She [topic-は] tears [subject-が] with-a-rush overflowed.
どっと is a mimetic adverb for a sudden rush or flood — things coming all at once. Here it modifies あふれる ('to overflow, to flood'), painting a vivid image of tears bursting forth. The construction shifts: 彼女は is the topic, but 涙が is the grammatical subject of あふれた.