The answer is
ENTERPRISE, a (famous) undertaking
♪You've known this puzzle for so long,
The tv series Star Trek debuted in 1966, over 50 years ago, and featured weekly episodes of space-faring adventures aboard the USS Starship Enterprise.
You know the rules, and so do I.
A full description's what I'm looking for
Yup, explaining all the parts of the puzzle is how we do it here. Ok, here goes ....
Of this vehicle for your entertainment
The starship Enterprise, the famous vehicle of the Star Trek series of shows and movies.
I'm just gonna break it into pieces,
gotta check you understand:
This is a "Riley" style riddle, so the answer will be described piecemeal.
I'm gonna end, you'll hear,
with a nice reward
PRISE, which we hear as "Prize", a nice reward. This, we're told, is the end of the word.
But to take it will take effort.
This might be cluing any number of things:
It might refer to the starship Enterprise being difficult to capture.
It might be that an enterprise is often something hazardous or difficult.
Or, probably intended here, is that this line goes with the prior two, because
"prise" means "to move or force, especially in an effort to get something open".
You'll find I gave my name
to my ancestor
The original Enterprise NCC-1701 from the original television series has gone on to give its name to various other incarnations, including the NCC-1701-C seen briefly in The Next Generation, and the NCC-1701-D, the featured vehicle itself of TNG. It also was used for the "prequel" series ENTERPRISE, both as the name of the show and as the name of the ship, the NX-01.
Also, a U.S. space shuttle was famously named Enterprise (OV-101) following a successful letter-writing campaign by fans of Star Trek to give it that name (its original planned name was Constitution).
And I start with an arrival.
ENTER. This, we're told, is the start of the word.
The sky's the limit, or so they say
A vague space reference, perhaps.
I keep on going, to the very last border
Space, the final frontier
You can argue who is best in charge,
A commentary, most likely, on the debate over who made the best captain - Kirk or Picard.