No. Check what each communication covers and which intended applicants still have requirements outstanding.
Track people as well as the shared contact
The FAQ recommends initial inclusion of the eligible spouse and unmarried children under 21 who intend to join. Each adds US$15,000 and a US$1 million gift; neither shared access nor one person’s completed response establishes everyone’s eligibility or progress. Build a row per person and a column per requirement, and fill it in from the official record rather than from memory of what was submitted.
The question this answers is not whether the family can see its messages but whether each intended person has an outstanding step, and those are unrelated. One relative's completed response tells you nothing about another's, and a shared mailbox showing no new mail tells you nothing about either. Two consequences follow.
An intended spouse or unmarried child under twenty-one who was not actually included at the outset is not included by virtue of living in the household, and adding someone later is its own question on its own terms. And each included person carries the published processing amount and the published gift amount, so a household reviewing its position should be reading a list of people rather than a single number. Confirm the list itself before confirming anything about its progress.