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Trying to make a model that predicts the stock market's total index by giving dollar price and inflation percentage as inputs, I know for sure, By changing the value of dollar price the stocks market will be affected ~10-20 days later!

Can a neural network model detect the effect of input in the future outputs? If yes, What models are the best fit for these situations?

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I'm assuming you mean inflation percentage? Any machine learning model for regression will work here, including even linear regression up to neural networks. There is no one model guaranteed to always give the best results for any given problem, so you will need to try a few and see what works.

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