What is the name for a person who raises turkeys?

Solution 1:

The word you are looking for is poulterer (there is an obsolete version, poulter), or perhaps poulteress. All are found in the OED in this sense.

Solution 2:

In Australia, the general term, depending on the level of involvement, would be Turkey breeder or Turkey producer.

Solution 3:

Poult (The American Livestock Breeds Conservancy)
"is a young domestic (not wild) turkey."
"Poultry are domesticated birds that are kept for meat or eggs including birds of the order Galliformes: chicken, turkey, natatorial (swimming) birds: duck and goose."

Poultry (Wikipedia)

If you want to be technically specific, it would be poulter.
Informally, you can say turkey farmer.

See also:
poultry (dictionary.reference.com)
mid-14c., from O.Fr. pouletrie "domestic fowl" (late 13c.), from poulet "young fowl" (see pullet). Poulterer (1630s) is a redundancy, but has largely ousted original poulter (c.1400), from O.Fr. pouletier "poulterer," with agent suffix -er. Poetic poulter's measure (1570s) is of fanciful origin.