There are a lot of companies that could do it, check with Luis and Rosie over at Toucan - http://www.tucanlimoservices.com/ they are popular and get consistently good reviews. You can check with Costa Rica Shuttle http://www.costaricashuttle.com/ another good company. Those are a good place to start.Edited: 18 September 2017, 19:35
If DB, you need to get to Sierpe, then take a boat, which goes twice daily, wet landing in DB. Your hotel in Osa might be able to reserve seats on the boat.
When are you travelling? Puerto Jimenez airstrip has been closed, will continue to be closed till Oct. (the first reopening date was supposed to be this past Aug., then became Oct.), so even if you wanted to fly a domestic airline there, you could not, only to Golfito, then a water taxi / ferry to Puerto Jinrnez. Or driving SJO to Puerto Jimenez. The DB airport should be currently open.