Can Fisher-Yates shuffle produce all playing card permutations?
You are right. With 128 bits of starting state, you can only generate at most 2128 different permutations. It doesn't matter how often you use this state (call Math.random()), the PRNG is deterministic after all.
Where the number of calls to Math.random() actually matter is when
- each call would draw some more entropy (e.g. from hardware random) into the system, instead of relying on the internal state that is initialised only once
- the entropy of a single call result is so low that you don't use the entire internal state over the run of the algorithm
In general, if a pseudorandom number generator admits fewer than 52 factorial different seeds, then there are some permutations that particular PRNG can't choose when it shuffles a 52-item list, and Fisher-Yates can't change that. (The set of permutations a particular PRNG can choose can be different from the set of permutations another PRNG can choose, even if both PRNGs are initialized with the same seed.) See also this question.
Note that although the Math.random algorithm used by V8 admits any of about 2^128 seeds at the time of this writing, no particular random number algorithm is mandated by the ECMAScript specification of Math.random, which states only that that method uses an "implementation-dependent algorithm or strategy" to generate random numbers (see ECMAScript sec. 20.2.2.27).
A PRNG's period can be extended with the Bays-Durham shuffle, which effectively increases that PRNG's state length (see Severin Pappadeux's answer). However, if you merely initialize the Bays-Durham table entries with outputs of the PRNG (rather than use the seed to initialize those entries), it will still be the case that that particular PRNG (which includes the way in which it initializes those entries and selects those table entries based on the random numbers it generates) can't choose more permutations than the number of possible seeds to initialize its original state, because there would be only one way to initialize the Bays-Durham entries for a given seed — unless, of course, the PRNG actually shuffles an exorbitant amount of lists, so many that it generates more random numbers without cycling than it otherwise would without the Bays-Durham shuffle.
For example, if the PRNG is 128 bits long, there are only 2^128 possible seeds, so there are only 2^128 ways to initialize the Bays-Durham shuffle, one for each seed, unless a seed longer than 128 bits extends to the Bays-Durham table entries and not just the PRNG's original state. (This is not to imply that the set of permutations that PRNG can choose is always the same no matter how it selects table entries in the Bays-Durham shuffle.)
EDIT (Aug. 7): Clarifications.
EDIT (Jan. 7, 2020): Edited.