You are submitting a tax return, not a W-2, which is just one form that goes into a tax return.
If your tax return is accepted (by the IRS? NYS? I don't know which one you are referring to), then it can be in as little as three weeks to receive a refund, if you are using direct deposit. However, there are a number of factors that might slow that down.
"after we respond to their request." What request are you referring to?
@louis878
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