A new missionary endeavor has prompted a "missionary start-up" series of questions. His situation is very typical and may be helpful to rehearse with all viewers of this blog.
"The Lord has called me to be a missionary. At this time, I am still working and earning money in the United States. I not ordained at this time.
"After sharing my burden with our local church, a mission agency was created by our local church to help facilitate the funding for my family's’ mission needs. The only step that was taken to create the mission agency was to open an account at our church’s bank in the name of [the] Agency. We also opened a sub-account under the agency in my own personal name. [The church's agency] received a Tax ID number for this account. [Since 2009, contributions] have been accumulating in the two accounts.
"No money was taken or used from either account until August of  when I took a survey trip. During my trip, I incurred expenses for taxi services, hotels, eating, obtaining a passport, and purchasing plane tickets. This money was taken from the sub-account in my personal name, most of which had been transferred from the “missions agency” account."
Were we required to report the income that was given in 2009 to the government in last years’ taxes?
No, as long as the church agency maintains complete control of the funds and their disbursement, you have no income until you are compensated for your personal services. In fact, to avoid potential confusion, if asked for my personal counsel I would likely advise you to close the personal sub-account under the church agency Tax ID. When the agency distributes personal living expense money to you, it should pay it to you for deposit into your own personal family account. At that point, these earnings must be reported to you and the IRS at year end on Form 1099-MISC or Form W-2 (depending on your employment or independent contractor status with the church agency).
How do we claim income in 2010?
As you have described it so far, you have no taxable income in 2010. It appears that the monies disbursed to you were in reimbursement for business expenses, not for compensation of your personal services. The church agency should maintain copies of substantiation of your business expenses to document that all of the disbursements were business travel costs.
Does the fact that the church [has not pursued formal incorporation] mean that the church cannot consider me [its] employee, legally speaking?
This should not make a difference. To my understanding and experience, virtually all state statutes recognize churches as corporations regardless of their formal status.
There are many other issues this new missionary will need to consider. A careful reading of past entries in this blog could probably be a good start. They include choices regarding ordination/licensure, employee vs. independent contractor status, financial management arrangements with the church agency, self-employment tax, and many other issues.