If my company is tax exempt can I avoid paying hotel taxes?

Maybe. It all depends on what agency you work for, in what state, and where you are traveling to. State government employees are generally exempt from hotel tax within their own state only, and each state has its own regulations for which of its state employees are exempt and how that is handled (up front or reclaim afterwards; only if paid with a government card, etc.). Each state also controls whether federal government employees are exempt. Only a dozen or so states offer this benefit and qualifications vary though all require that travel be for official business and paid with a government credit card. Even within the states offering the benefit there are hotels that don’t honor it.