I just took a gander as I am planning to head out to Houston to visit family.
This site clearly states that you need a permit:
It says you need one "to conduct sidewalk performances in the theater/entertainment district"
(Theater/entertainment district shall mean the area including to the mid-point of and bounded by Preston Street on the north, Dallas Street on the south, Milam Street on the east, and Interstate Highway 45 on the west.)
Presumably other districts are not regulated. Did not find "sidewalk performances" anywhere else in the municipal code.
(Bear in mind, this is from a paralegal candidate in Canada; I'd still call city hall, ask for the bylaw or licensing department and ask them.)