US citizen forming a TX-based LLC but doing business abroad.
My LLC's primary business will be consulting for web development clients on a time basis.
You may register an LLC in any state, including Texas, of course, listing either a physical U.S. or Non-U.S address. LLC is a pass-through entity, meaning the profit is passed-through to the members' without being taxed on company level. The members then pay personal income taxes to the state/country they reside (U.S. citizens also must file federal tax return).
In your specific case, it would be a good suggestion to consult with a licensed accountant to understand exactly how your taxes will be applied and what write-off benefits you can utilize by having the LLC registered in your home state of Texas. We have an article discussing taxation of foreigners in the U.S., and there is a brief section discussing taxation of U.S. businesses owned by U.S. person(s) living abroad which I invite you to review.