Is it possible to guarantee a SHA-256 hash will begin with a certain 3 bytes?
Cryptographic Hash functions are deterministic random functions. Deterministic means the same input the same output and random in a sense mean that the output is unpredictable. We expect that the hash functions have avalanche effect that whenever one bit complemented from the input, each of the output bits flips with a 50% probability. 

If it is possible to find a specific first 3-byte faster than the hashing random inputs this actually means that someone found a weakness in the hash function. There is no known such weakness in the SHA-256. 

Note: By the design of the SHA-256 (or the family), even there is a non-random method for the first 3-byte, with high probability, the other bytes will have, too.