Community Wiki, as it's unclear to me why @JaapJorisVens deleted his answer given that it appears to contain the necessary clues:
The carts are special carts we have made that contain 1000 selectable games. It is worth mentioning that the carts are actually adapters - essentially Game Genies - which do nothing unless you have an original NES Tetris cart plugged in.
Each of the 1000 games corresponds to a unique random seed which initializes the RNG for each game. The seed is iterated once per piece to produce the sequence. ...
Part of the RNG algorithm in the cart makes use of fairly large tables ... Each year we shuffle the contents of these tables to make a new cart.
- the original Tetris is used;
- a competition-specific cartridge acts as a passthrough to replace Tetris's random number generator; and
- specifically, that passthrough device spots when Tetris is performing its new-piece sequence and at that point inserts the next random number.
The NES itself doesn't inherently generate random numbers, that's a per-game process, implemented however each game sees fit. For the purposes of this competition they've engineered an intermediary that substitutes its own random numbers, generating them exactly on-demand.
Per other parts of Jaap's answer:
The math involved in 'iterating' the seed is fairly detailed, involving a LFSR (as is used in the standard NES cart), a few looping counters, and some lookup tables. Just before each game, a random number between 0 and 999 is chosen and players are told which game to select on the cart. After a game is played, it is never re-played.
So both the new implementation and the original are linear feedback shift registers ('LFSR's), which are shift registers in which the bit going out is exclusive-ORd onto some other combination of bits in the current state. Through that means an n-bit register can iterate through 2n − 1 states in a fixed sequence that is sufficiently oblique that a human might interpret it as random.
That said, it sounds like the passthrough device goes further in using the LFSR to index random number tables, so it's likely that the LFSR is being used just because it's easier to throw together than a classic ripple counter.
Quite probably the original just uses the output of its software LFSR directly.