Can an agent accept a bonus directly from a client without the approval of their broker?

No. A sales agent can only receive compensation, including bonuses, with the written consent of their sponsoring broker, according to The Real Estate License Act (TRELA) §1101.651(b) and (c) and TREC Rule §535.3 and only if such bonus is accounted for in the buyer/tenant representation agreement.

For example, if a builder offers a cash bonus or a car to agents in return for completed transactions, the sponsoring broker must approve the compensation before the agent can accept it. If the sales agent changes brokers, the sponsoring broker of the agent at the time the agent became entitled to the compensation can approve the payment. Similarly, if the buyer/tenant representation does not contemplate the payment of such bonus, the client would have to agree to amend such agreement to reflect the bonus. Keep in mind that NAR has stated that there needs to be a reason for amending the buyer/tenant representation to increase the amount you are paid such as additional work completed for the buyer.