No, online based businesses doesn t need a license in Texas but you do need a to file a assumed name certificate at your local courthouse. Since you have a e-commerce site you need to collect Texas sales tax from Texas residents, it s called a sales tax permit. You can get the assumed name certificate at your local courthouse and fill out the form for sales tax permit on the Texas Secretory of State s web site.
I don t think you need a license to sell online, no. You need all the documents you wouldnt need for a real-world store though.