Yes, Shallots can be grown here but don't expect large size bulbs compared to what is purchased from the grocery store.
Shallots are going to multiply and grow into bulbs about the same size as what you are planting. In South Texas they are considered to be a fall crop, planting in late fall (Oct 15 - Nov 15).
See the information at this PLANTanswers Web site:
Description - Considered to be the gourmet member of the onion family due to its mild, delicate flavor, shallots are very similar to green onions, but develop in clusters of small bulbs rather than as individual onions. Shallot bulbs are small, generally elongated, and have a distinctively different flavor and odor as compared to onions.
Culture - Grown very similar to onions. Planting is usually done in the fall or early spring. Plant small bulbs about six inches apart leaving the growing point exposed above the surface of the ground. Maintain adequate fertility and moisture conditions during growth of the plants. Harvest in early summer when the leaves turn brown. Save some bulbs for the next seasons garden.
Availability - Shallots are not commonly available in all supermarkets and grocery stores, but are becoming more popular. Shallots are not grown commercially in Texas. Some production occurs in the southeast and in California.
Selection - Look for bulbs that are firm and free of rot and bruises. Shallots with yellow skin have a strong flavor. If green shallots are desired, select those that have strong, disease free tops and healthy appearing stalks.
Storage - Dry shallot bulbs store well in ventilated, cool but dry areas. Green shallots should be stored in the refrigerator and used within about two weeks.
Nutrition Information - Shallots are a source of potassium and are low in sodium. A 3-1/2 ounce serving contains 72 calories. Preparations - You may use shallots whenever green onions are called for. When ready to cook, cut off tops and roots, wash white part, separating layers and letting water run between them to remove all traces of dirt. Cook covered, in chicken broth or water until tender, 15 to 20 minutes; drain. Season with salt, pepper and butter or margarine.
Here is a link with more information on growing Shallots.
Our local Independent nurseries usually carry bulbs for sale.
You can also purchase adapted bulbs from http://www.johnnyseeds.com/c-46-shallots.aspx.
Extension Horticulturist- Bexar County
Texas A&M AgriLife Extension Service
3355 Cherry Ridge, Suite 212
San Antonio, TX 78230 210-467-6575