Xanterra, the ground operator at several of the busiest National Parks, announced that it will refund deposits for anyone who has prepaid for a stay. Xanterra will also offer prorated refunds to people already in a park forced to leave early. Xanterra says it has no responsibility for airfares and any other trip payments other than for its own accommodations.
That's probably about the model that most ground operators and tour operators will adopt. Operators of package tours including National Park visits will almost surely try to get you to reschedule rather than ask for a cash refund, but you'll probably get a refund if you ask. If airfare isn't included in the package, however, the operator will not refund tickets or compensate for change penalties.
According to Quotewright's John Cook, my go-to travel insurance maven, policies do not include government shutdown as a "named peril" or "covered reason" for cancellation or interruption. Except in the case of cancel-for-any-reason policies, you could recover only what you could get from the airline, tour operator, or accommodations operator.
I'm still waiting for responses from major airlines about any easing of their ticket-change penalties and fees. I'll post as soon as I hear anything.