The City of Lubbock requires a permit before code-regulated building work starts, and it counts any interior change to an existing home as remodeling, but the list it gives names structural, electrical, plumbing and mechanical work, not insulation. So I won't guess whether your job needs one. When it does, we get the permit from the city where the house sits and take care of the paperwork, and if we aren't sure, we call that city's building office and ask before we start. You can ask too, since the city's Building Inspections office takes questions about specific work at 806-775-2087 or 806-775-3155.
The city also says building contractors and remodelers don't need a Texas state license, but they do have to register with that same office and show proof of commercial general liability insurance or a compliance bond, so registration is a fair thing to ask any contractor about. We do the same foam work in Wolfforth, Slaton, Levelland, Plainview, Littlefield and Brownfield, and the calls there sound a lot like Lubbock's, older houses with tired attics and rooms that never feel right.