the word simply means different things to different people. Basically I assume you want to be self sufficient for some period of time, you want to have a vehcile that will transport yourself and whomever else in relative comfort, you want the vehicle to have certain capabilities, and you want to bring along enough resources to make all of that happen. Obvioulsy you want a dependable vehicle that is appropriate for the geography you encounter...etc...etc.
I went on a 9 month expedition one time....it involved a lot of sun and sand...and (first gulf war)...really the only "expedition" I've ever been on. I slept in a tent for about 8.5 months, sometimes in the back of my truck, I had all the water and MRE's I could eat and drink, I had an outhouse at times and then a shovel at times for a bathroom, I had an outhouse shower at times and then nothing at times. I had one hot meal a day for a while, and then MRE's forr the rest. Hard to get clean hard to stay clean, I had sand and dust in everything and everywhere, I had more flies than I thought would have ever existed. A simple hot or warmed up portion of food was a luxery at times, somethign to drink other than water was a luxery too, at times. I washed clothes in a bucket of water, We went days at times without seeing another animal or living creature (in the desert) we drove many ...many hours /days in the sand and where there was absolutly nothing in a 360 radius for miles and miles. Where you would certainly die if your vehicle became stranded and you ran out of water.
My obersvations on the expdition I describe....the capability of the vehicle is critical (we got stuck many times) but there were a lot of use together with various types of vehicles so we helped each other out. Large heavy vehicles suck in the desert (loose sand) and those with trailers do even worse. Hot food is great, options dealing with food and liquid are great. GPS is a great thing if you have it, otherwise best to know how to read a map. Communicaiton is great..of course military stuff is different that civilan but I would think somethign other than cell phone would be smart to have and know how to use (ham radio). Learn how to be self sufficient, carry spares, carry only what you need and look for clothes and other devices that could serve dual roles, know your supply points, ask the "locals" if there are any, don't go by yourself, know your vehicle and have common tools, know how to work / use all the devices tools before you leave, practice to make sure. travel light as possible, get organized, figure out good storage palces or build some, do some bain-storming well before you plan on leaving, ask those who have gone before for tips / advice.
Obviously you need food, water, the ability to prepare food, the ability to store water & fuel, resources relaed to camping, potentially a heat source, and understanding of field sanitation, understanding of land navigation, communication equipment, maps, clothing, maybe even the abilty to speak other languages, and an understanding of field hygene.
The best solution is the ability to live with the least amount of resources that you potentially can. Would probally be a good idea to do some practice runs to make sure all teh equipment functions as intended and you know how to setup everything in all weather conditions and even in total darkness.
I assume an expidition is something athat is longer than a week or two camping out. The people who are good at expiditons are probally the ones who can live with the least amount of luxery items.
There is no one solution to the vehicle question...all depends on where you plan to go and what you want to carry. I think a person who could tolerate to live pretty minimal probally could get buy in a 4wheel drive tacoma with a camper shell, all depends on how you want to live, how much you are willing to tolerate and what you consider is a good time. of course if you are going carry others with you...what are they willing to put up with and live without.
Military field units like the field artillery unit I was part off...carry their whole world in the vehciles... but we also had re-supply and probally could have lived self sufficient for a few weeks at the time given our resources on the vehcile at the time (way back in 90/91)... dealing with insects is also fun...appropriate headgear with an insect barrier is great depending on where you are. You can tolerate insects on your body etc...but really hard to deal with them flying around and biting you on the face. Water and liquids attract insects.
I think a FJ80 in good running condition would probally serve as a very capable expedition vehicle for most, because I suspect that most people will not be to far away from civilization for that long of a period. Select the right tires as appropriate, winch, bumpers, etc would be a plus, slidders, guards..