Can a real estate agent buy the property they have listed?

Yes, if certain conditions are fulfilled. IT is NOT an optimal situation for either party. In most US states the agent must have a signed disclosure that the seller is aware that the buyer (in this case the listing agent) is a licensed real estate agent and is not representing the seller's interests in the case of an agent purchasing on his own behalf.

Most ethical agents regard this as a gray area, because the agents will have so much more information than a non-agent seller. Many agents will not do this type of transaction. If the seller closes and then concludes he did not get a fair price; chances are the seller will complain to the real estate commission. This will result in inquiries and a time consuming process. The best practices solution would be to cancel the listing agreement and have the seller get another agent to represent his interests.