I'm looking to outfit my Ubuntu laptop with a satellite card so that I can access the Internet from just about any place on planet Earth.
I searched on Google "satellite network card" and was pretty disappointed by what I found.
Do such NICs simply not exist, or not exist at the consumer level?
I cannot understand why you are fixated on "cards". The days of almost all laptops having PCMCIA cards are long gone, and the ExpressCard slots that replaced them are getting rarer. Also the latest, greatest laptop expansion card interface has been far less stable and less long lasting than e.g. ethernet, USB, or WiFi.
why are they not turning up in those search results?
Because you want a "satellite modem". A "satellite modem hotspot" might be an even better option, since you can use it for multiple devices and for any laptops you get in the future. You may also find some leads in the more obvious "mobile satellite internet".
"Satellite network card" has about as much meaning as "DSL network card" would, and mostly would be dominated by the popular topic of satellite TV receivers.
Note that in general you will usually have better search results if you search for what you are trying to accomplish, rather than what you think you need to do it. Even "access the internet from anywhere on Earth" has a few on the first page. Even if that doesn't get you started right out of the gate it will at least give you some ideas of better things to start searching for.
how much "coverage" do they actually give you
Once you browse through the above results, you will be able to answer this question on your own by researching whatever vendors are available. Even if I had a full list it would not be a good answer here anyways, as it will undoubtedly change; so just start with whatever the current search results are and go from there.
what kind of special service plan do you need to leverage them?
Same as above. Each vendor will have their own plan details, and these details are likely to change over time, so check the vendors' sites for their latest offerings.
In my experience, satellite internet access is generally expensive, both the hardware and the service plans. What you may want to do instead is get a global 3G/4G cellular hotspot (or a global phone with a tethering option) with a SIM card slot, buying SIMs as you travel, for coverage in most populated areas, optionally a cell signal booster, then if you have a satellite modem see if you can find a plan with no access fee where you just pay for data used, which you can use as a backup. Personally, I would definitely not use a satellite link in areas where other services were readily available; it is more of a backup plan.