I'm pretty sure a friend of mine who went several years ago purchased one over the telephone, I'm not positive though...but I just remembered I'll be in Houston next week so I'll just pick one up then. If I remember correctly they let you set the dates of the liscense at the time of purchase.
:I have never looked into buying a license online, so my answer is not 100%. But I do not think you can, as far as I can tell they need a photo ID to sell you a license in Texas. I do not know how they can get around that online. :Most sporting good stores, even Wal-Mart can take care of you when you are here.