One of my friends is an American citizen, but her husband and kids are not (B-2 visas). Their I-94s expire on May 15th, so they are applying for Adjustment of Status with I-485 and I-130 forms. I know that their period of authorized stay extends after May 15th once their forms are received and their case is "pending" at USCIS. But is it legal for her husband and kids to stay in the United States in the time period between the mailing of the form (before May 15th), and the receipt of the form by USCIS after May 15th?
They are not subject to deportation while their Adjustment of Status application (I-485) is pending. An application is considered filed when it is officially received by USCIS, not when it is mailed. (Of course, they wouldn't know the date it was officially received, until they receive the I-485 receipt, there will be uncertainty about when exactly it was filed. But when they do receive their I-485 receipts, it will be retroactively clarified.)
If their I-485s are received on or before May 15 (if that is the expiration date of their I-94s), they were not deportable at any point. If their I-485s are received after May 15, there will be a period of time (from after their I-94 expires until their I-485 is received) when they will be technically deportable, but it is extremely unlikely that they would be caught and put into deportation proceedings during a short overstay. (And even if they are put into removal proceedings, Adjustment of Status can still be filed in removal proceedings, though it's more complicated.)
As for the effect on their Adjustment of Status application, it makes no difference whether the application is received before or after May 15. Although being out of status at the time of filing makes one ineligible for Adjustment of Status in most categories (see 7 USCIS-PM B.3), it does not apply in the Immediate Relative category (spouse, parent, or unmarried under-21 child of US citizen), as is the case with the applicants here. See also 7 USCIS-PM B.8(B):
Certain adjustment bars do not apply to an immediate relative, including the spouse or child (unmarried and under 21 years old) of a U.S. citizen, and the parent of a U.S. citizen older than 21.
An adjustment applicant applying as an immediate relative may be eligible to adjust status even if:
The applicant is not in lawful immigration status on the date he or she files the adjustment application;
The applicant has ever failed to continuously maintain a lawful status since entry into the United States;
This applies no matter how long the person was out of status. Even if someone's status expired years ago, if they are in the Immediate Relative category and still in the US, they are just as eligible for Adjustment of Status as someone who is in status.