non-repeating random numbers

You don't want random numbers at all. You want exactly the numbers 0 to N-1, in random order.

Simply filling the array and shuffling should be very quick. A proper Fisher-Yates shuffle is O(n), so an array of 100 million should take well under a second in C or even Java, slightly slower in a higher-level language like Python.

You only have to generate N-1 random numbers to do the shuffle (maybe up to 1.3N if you use rejection sampling to get perfect uniformity), so the speed will depend largely on how fast your RNG is.

You'll never need to look up whether a number has already be generated; that will deadly be slow no matter which algorithm you use, especially toward the end of the run.

If you need slightly fewer than N total numbers, fill the array from 0 to N-1, then just abort the shuffle early and take the partial result. Only if the amount of numbers you need is very small compared to their range should you consider the generate-and-check-for-dups approach. In that case Bob Floyd's algorithm might be good.

As an alternative you could use an appropriately sized block cypher. Use the block cypher to encrypt the numbers 0, 1, 2, ... and you will get a series of non-repeating random numbers out. Exactly what series will depend on the key you use. They are guaranteed not to repeat, because a block cypher is a reversible permutation.

For 64 bit numbers use DES, for 32 bit use Hasty Pudding (which allows a large range of block sizes) or write your own simple Feistel cypher. Assuming that security is not a big issue for this, then writing your own is possible.