戻ってくると、千波が我先にと駆け寄った。
once we came back, chinami was the first to run up to (us).
how does this sentence change without the と after 我先に? Doesn't seem to quotatitive or conditional.
thank you
This と is "the optional と" that attaches certain adverbs. In your case, dropping と does not change the meaning of the sentence.
See:
我先に【われさきに】 is a set phrase meaning "striving to be the first". It does not necessarily mean Chinami was actually the first one.