No palms of any kind can be shipped to HI.

The rest must be declared, inspected, and forms filled out from the plant quarantine folks...usually with a fee associated.

Your citrus and papaya will be heavily inspected...probably the avocado, too.

Keeping disease/pests/animals that aren't on the islands already from getting onto the islands is a huge concern that they monitor closely.

http://hdoa.hawaii.gov/pi/pq/travel-shipping-information/

That link is the Hawaii Plant Inspection folks. Here is a handy page which lists how to bring in plants from various places: http://hdoa.hawaii.gov/pi/pq/import-program/plant-guidelines/ You may have to do a copy and paste into your browser window to get to the links since I'm not sure how to put them into the page here.

For some reason, they won't allow palms from the mainland, but you can bring them in from foreign countries if you have the proper paperwork.

Shipping plants in usually requires them to have no soil so either bare root or some sort of soil-less medium is used on the roots.

All of those plants you've mentioned are easily found in Hawaii in all sorts of varieties. A lot of folks get trees from Plant It Hawaii: http://plantithawaii.com/ They are a plant wholesaler, but they have big sales twice a year where folks can just get one or two trees. There are also big plant sales by the Big Island Association of Nurserymen (BIAN) sales. Those sales have all sorts of plants. If you need more banana plants, folks will give them to you when theirs make too many keiki. Papaya sprout when you are eating a papaya outside and just toss the seeds on the ground. Papaya are also a fairly short lived tree/plant, so it might be time for a new one when you move.

If it gets too expensive or difficult to ship the trees, you could perhaps bring cuttings and start new avocado, grapefruit and lemon trees from the cuttings.

Here is a link that might be useful: Hawaii Plant Import website