It was all online, neither of us were asked for any proof. I am in Colorado so we use a state health marketplace instead of the federal one but I assume it’s pretty much the same process. It asked for my husband’s marital status, immigration status (adjustment to LPR), US income and SSN. The marketplace then gave us a list of insurance companies and policies to choose from.
Generally in our AOS process, the only documents we’re ever had to show to prove immigration status are the I-94, marriage license and I-797 (which is the NOA1 for AOS). However I didn’t have to show any of these for insurance, the healthcare marketplace verifies identity through social security numbers.