How can I legally visit the United States Minor Outlying Islands in the Pacific?

I can answer for Palmyra Atoll, as I happend to have come across this recently.

From time to time, the US Fish and Wildlife Service seeks volunteers to work on Palmyra Atoll. The last call for volunteers went out in May, and it's a bit of a commitment: "Volunteers will be expected to work 8 hours a day 6/days per week for the entire ~3-4 months on Palmyra, without any chance of leaving the atoll." So you could look out for future volunteer opportunities, if that appeals to you.

If you'd like a shorter trip, FWS describes the methods for authorized visits:

There are four ways the public may gain access to the refuge:

• Working for, contracting with, or volunteering for The Nature Conservancy or Fish and Wildlife Service
• Conducting scientific research via Fish and Wildlife Service Special Use Permits
• Invitation through The Nature Conservancy sponsored donor trip
• Visitation by private recreational sailboat or motorboat

Donating a bunch of money to the Nature Conservancy could do the trick, and I'm sure their development department would be happy to discuss the amounts required. Or you could arrange a boat to take you out there (or learn to sail and get your own boat). That page further describes the requirements for private visits (limited to 7 days); you'd have to get in touch with FWS staff to determine when they're able to accommodate a visit.

A final note from that NPR story:

And a tip — a final field note — for those who do get to the island: Before you go to bed, shake out the sheets. Often, brown spiders the size of silver dollars live in the folds of the bed spreads.

Join or sponsor an Amateur Radio expedition.

Many Amateur Radio enthusiasts like to collect radio contacts and postcards confirming these contacts (called QSL cards) from rare and distant locations (called DX).

The richer hobbyists will sponsor or join a DX-pedition to some of the more exotic, uninhabited, locations that are rarely heard on the air. Inhabited places already have Amateur radio operators, as it is widely accepted as a hobby or public service. Often these DX-peditions are timed to coincide with specific, well known yearly contests where they compete at generating the most ham radio contacts or the highest contest score.

An argument that is often put forward when advocating for permits, etc. is that these expeditions (and the related contests) have value as emergency preparedness training exercises. Organizing an effort to take equipment, wire, antennas, portable or improvised towers, and portable power generators out into the middle of nowhere with no infrastructure and set up contact with the world can yield some insights into readiness and operations after a hurricane, earthquake, or other event in civilized areas when ordinary communications has been disrupted. Those participating in a contest from their comfy air-conditioned home may find special categories or extra points for using low power to make contacts or for disconnecting from commercial electricity and using non-polluting solar/natural power sources. The overlap with actual emergency ops isn't perfect, of course, but maybe it doesn't need to be.

As for these specific islands, I found previous efforts by Googling a location+" dxpedition"

Palmyra Atoll - K5P

Johnston Island - K3J Notes military has been busy clearing land mines and chemical weapons....

Wake Atoll - K9W

Reading through these stories, when money is discussed you will find most of the financing comes from the groups themselves. Although it is possible that some hobbyists may include a few dollars with their request for a QSL card, and some of these efforts yield tens of thousands of contacts, not everyone wants a card. Also, most countries' Amateur Radio regulations prohibit soliciting for funds directly on the air.

I would imagine that enough resources, skills, money, and networking would yield a chance to be part of a group visiting one of these places.