Efficient data structure for storing integers in a range?
Since you know you're going to have to deal with all $2^{32}$ values eventually, you're going to need at least $2^{32}$ bits of memory, one for each value. The pigeonhole principle means that there's no possible way to store all the information you need with fewer bits than this.

So I recommend a straightforward bitmap. In other words, a simple array of bits. When a new value comes in, see if the corresponding bit is on or not; if it's off, flip it on. This takes the minimum possible amount of space, and is extremely fast (since all you have to do is index into an array).

P.S. You need $2^{32}$ bits specifically because the numbers could appear in any order. This means that there are $2^{2^{32}}$ possible states the program could be in, one for every possible combination of seen and not-seen values. Representing this many states will always take at least $2^{32}$ bits. If you knew the numbers would always come in increasing order, on the other hand, there'd only be $2^{32}$ states to distinguish between (since all you care about is the highest value seen so far), so you'd need a minimum of $32$ bits.