It’s easy to get a ride from Austin back to Dripping Springs on the fly. Heading to Austin, or elsewhere, you might look into booking in advance. https://help.lyft.com/hc/en-us/articles/115013078668-Scheduled-rides-for-passengers#how
Another option is booking a charter through Chariot. They use different vehicles based on the size of your group.
Cynthia’s Limo Service. They also have different vehicle sizes. I’ve personally used this company for Airport drop off & pickup for years. You can book online or give them a call.