They could not, because he does not own those businesses outright. Doing so would violate the other owners rights. They can seize his shares in SpaceX, but not SpaceX property.
But the end result is always the same. You can either voluntarily liquidate and pay your judgements or you can have someone else do it for you.
I have not done an analysis on exactly which of Musk's private companies are jointly owned, and how the ownership is structured. It's a comment on hn for crying out loud.
The only question is regarding which entity is liable. If a judgement were entered against SpaceX, for example, then absolutely rockets and real property owned by SpaceX can be directly seized.
Without looking, I'm certain there are at least a few companies owned by Musk which could have their assets taken directly.
They could contact the broker holding his shares and restrain them, order them sold, and collect the proceeds.
They could seize and sell at auction the real property of any of his businesses. SpaceX rockets? Office chairs? Servers? Anything.
All of this enforcement costs money. Twitter would account for the money spent seizing assets and charge those expenses to Musk as well.
Assets seized in this way (including stock) are often sold for much less than they're ideally worth. It's Musk's problem. Twitter would be empowered to seize seize seize until they've raised enough cash to cover the judgement plus expenses.