In this quotation:
Strange counterfactuals emerge in Vaughan’s tale, as a result of the very real disparities we live with every day. (Context)
What does "live with every day" mean? Can I use another preposition here? (live on/in every day)
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To “live with something” means to “accept something” or to “abide something”.
live with something (ACCEPT) to experience and accept an unpleasant event, decision, or situation: When you get arthritis at your age, it’s just something you just have to live with. —Cambridge Dictionary
The words every day can be deleted without changing the meaning of the sentence. They just mean the situation is continuous and unlikely to change.
“Live with” is a phrasal verb. If you delete “with”, it changes the meaning of the sentence.
Finally, if you delete “every day,” you’ll notice that the sentence ends with “live with” — a preposition at the end of the sentence. This is a good example of when it’s perfectly fine to end a sentence with a preposition. The writer may have used “every day” to avoid that occurring.