Random number generator with freely chosen period

I want a simple (non-cryptographic) random number generation algorithm where I can freely choose the period.

One candidate would be a special instance of LCG:

X(n+1) = (aX(n)+c) mod m (m,c relatively prime; (a-1) divisible by all prime factors of m and also divisible by 4 if m is).

This has period m and does not restrict possible values of m.

I intend to use this RNG to create a permutation of an array by generating indices into it. I tried the LCG and it might be OK. However, it may not be "random enough" in that distances between adjacent outputs have very few possible values (i.e, plotting x(n) vs n gives a wrapped line). The arrays I want to index into have some structure that has to do with this distance and I want to avoid potential issues with this.

Of course, I could use any good PRNG to shuffle (using e.g. Fisher–Yates) an array [1,..., m]. But I don't want to have to store this array of indices. Is there some way to capture the permuted indices directly in an algorithm?

I don't really mind the method ending up biased w.r.t choice of RNG seed. Only the period matters and the permuted sequence (for a given seed) being reasonably random.