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Is it statistically sound to dummy code a continuous variable (effect modifier) into quartiles and compare the odds ratios of an IV vs. DV in Q1 and IV vs. DV in Q4 in logistic regression? If so, how would I quantify a difference?

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Categorizing continuous variables is often done but is problematic. It loses information and involves the notion that something special happens at the boundaries between levels, this is usually not substantively sensible.

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Of course it is possible, you just select then some baseline category to which other categories are compared.

You can return value of logit or raw class probability from the logistic regression.

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