Hi R.. I'm a Child Passenger Safety Technician and there are a couple of solutions to your dilemma.
First to answer the question, no, you are not required by law to carry a child restraint onto a plane.
Now to address other dynamics. Children under 4 years and 40 pounds are recommended to be in an airline approved child restraint per the FAA, because the airline seatbelt does a very poor job of protecting them in an accident. The lap belt can rise up and injure their little tummies, or they can even get ejected, in severe turbulence.
To correct a common misperception, this is not an issue of "if the airplane crashes, we're all dead anyway so why does a car seat matter?"
The facts: MOST airline crashes are survivable! Most occur at take off and landing and for the most part, properly restrained passengers survive them. Also, in turbulence, proper restraint makes a HUGE difference.
Because you will be taking a taxi, I strongly advise you to have a child restraint for your 2 year old and a backless booster for your 8 year old. You don't need anything big or bulky. You can purchase the Cosco High-Back harnessed booster at Target for 40 dollars. It's a pretty bare bones seat, BUT it's also quite light and easy to port around. You can strap it to your luggage.
You will find that a properly restrained child on an airplane is more likely to be quiet and well behaved and less likely to try to run around or kick other passengers' seats. If nothing else, the person sitting in front of you will thank you for bringing a child seat!
Now, if in the end you decide NOT to take a taxi at your desitination, the CARES harness will suit you nicely on the plane. It's a four point harness that attaches to the lapbelt and keeps the child safely restrained.