Most adcoms can access your DAT scores 2-3 days after you take the exam simply by using their A-Client. I think that quite a few adcoms have no idea how to use the A-Client, though, or simply don't use it at all, so that may not be an option.
Regardless, they should have the scores within 2 weeks or so, depending on the time of the year you take the exam (how busy they are sending out DAT scores/how busy schools are filing your information).