Maybe:
Writing style of the ransom notes looked familiar to them?
Something suspicious about Alice's new will?
William insisting the bottle of chloroform in the house was for chickens, while Anna insisted it was for cleaning materials, then that bottle disappearing when the FBI asked for it? Even more damningly, the FBI microphones picking up Anna telling William she had got rid of it?
Anna refusing to let the trash collectors into the basement to collect it as usual, the week Alice disappeared?
Perhaps the family knew, or found out, something about Alice's friends*
or the 'squab recipe Russians' William went to meet that day?
Or ( and this is likely, I think) they found out Anna had told them a pack of lies about her background, maybe from genuine Russian aristocrats who came forward when the story hit the headlines?