After much searching and prayer, we found an organization that, we feel, will give us the freedom and support we need to follow God's call. The organization is called Global Outreach Mission. Most missions organizations either want you to be members of a specific denomination or have their own missions they want to staff. Since we are looking to do work with a specific Vineyard church, our options for a sending agency became very few, very quickly.
Global's focus is on enabling individuals to follow God's unique call on their lives. This means that they send missionaries and support them, but they don't dictate where those missionaries will go or what they will do. There are some requirements, like you have to be joining up with an organization so you have a support structure around you, but otherwise we are able to do whatever work God is calling us to do.