Business/screenprinting
A Brief Digression on the Business of Printing T-Shirts
First of all, the designs aren't "ours" (we wish they were!); they either belong to (or are licensed from the design owner by) the company from which we buy the item bearing the design.
If that sentence is confusing, try this as an example: Pink Floyd owns the "rainbow-prism" design from their album The Dark Side of the Moon. Liquid Blue and Not Fade Away (part of SunDog) have both licensed the right to make t-shirts featuring that design. We look through suppliers' catalogs, decide which items we would like to carry, display those items in our webstore, and purchase stock from Liquid Blue or SunDog as needed (see above). At no point were we involved in creating the art or making the shirt.
Second, most if not all of our suppliers have their own printing facilities dedicated to printing their designs on their shirts. Generally, they are not interested in receiving unsolicited designs. They usually get their designs either from well-known semi-commercial artists such as Ted Blaylock or Richard Biffle (to name but two out of dozens whose work we carry) or from commercial art licensing agencies.
If you are an artist with a good selection of designs, and the designs are attractive and of general interest, you may want to contact a Licensing Agency about carrying your work. (Note: I know next to nothing about this angle of the business; talk to someone who knows considerably more about it before you sign anything.)
If you just have one design, or the designs are only of interest to a small group (club, church group, family, etc.), or if you have the funds to go this route and want results quickly, you might go looking for a screenprinting company in your area. (Note that some suppliers have their tie-dye designs available without printing on them, so you can use them as a background for your design -- SunDog and Liquid Blue come to mind; there may be others.) The drawbacks are (1) high up-front investment, I would think at least 3 digits, more likely 4; (2) you will have to arrange for distribution separately, if you are trying to make money selling the fruits of your labor. (If you are just printing for a small group, #2 may not be an issue.)