I'm also not aware of a minibus service that runs from HH to Chumphon.
As chopsticks says, there are regular buses there and they'll leave from the southern bus station which is located on Petchkasem Road south of Market Village.
Have a look at the link. The telephone number also works, but have a Thai speaker asking the questions.http://www.tourismhuahin.com/busses-huahin.php
I don't think you'll get much information on Chumphon from the HH-BKK bus station in the centre of town as they don't run the route.
Buses will take approximately 3 hours. Alternatively, there are trains which will take about 4 hourshttp://www.tourismhuahin.com/trains-huahin.php
If you take a bus, a word of warning. Make sure that it stops in the centre of Chumphon. Some of them only drop off at the bus station on the main road just south of the right turn to Ranong/Phuket. In that instance, you'll be around 6kms out of Chumphon town centre and whilst there are motorcycle taxis into town, it's pretty inconvenient, especially after dark.