Much has changed in regard to termite inspections since I began practicing 27 years ago. It used to be standard that the Seller would provide a clean termite inspection report at closing. The trend seems to be moving away from contractually requiring a Seller to provide one at closing.
HUD, FHA and VA which all previously mandated a termite inspection are now a mixed bag depending on the type of property you are buying, where the property sits on the Termite Probability Map and whether the appraiser notes a problem on his report.
Termites and other "wood-destroying pests" can be a serious problem for home owners, buyers and sellers. My advice to buyers and sellers is to get the inspection. The significant risk greatly outweighs the relatively small cost of the inspection.
Termite inspections are not typically included in the home inspection. A separate licensed termite inspector should be ordered.