How can missionaries and missionary organizations raise funds Biblically?
There is Biblical fundraising and there is unbiblical fundraising, and the way most missionary organizations and missionaries raise funds today is unbiblical, even Satanic (see below). Here is what the Bible says about fundraising.
1. Raising funds for Christians in need is Biblical.
When Christians in Judea were in need during a famine, Paul raised funds for them from Christians in other regions: "Now concerning the collection for the saints, as I have given orders to the churches of Galatia, so you must do also" (1 Corinthians 16:1).
2. Telling rich Christians to donate everything except enough to live on is Biblical.
"For I do not mean that others should be eased and you burdened; but by an equality, that now at this time your abundance may supply their lack, that their abundance also may supply your lack - that there may be equality" (2 Corinthians 8:13-14).
3. Raising funds from poor Christians to benefit wealthier Christians is unbiblical.
See 2 Corinthians 8:13-14 above.
4. Accepting donations from begrudging donors is unbiblical.
"So let each one give as he purposes in his heart, not grudgingly or of necessity; for God loves a cheerful giver" (2 Corinthians 9:7). Begrudging givers should be advised to pray to God until He changes their heart and fills it with joy, and then give cheerfully.
5. 100% of the funds raised must reach those for whom the funds was raised.
To ensure that 100% of the funds raised for the Christians in need in Judea reached them, the only people who even touched those funds on their way to Jerusalem were the donors' own representatives: "And when I come, whomever you approve by your letters I will send to bear your gift to Jerusalem" (1 Corinthians 16:3).
6. The fundraiser keeping a portion of the raised funds is Satanic.
But a certain man named Ananias, with Sapphira his wife, sold a possession. And he kept back part of the proceeds, his wife also being aware of it, and brought a certain part and laid it at the apostles’ feet. But Peter said, “Ananias, why has Satan filled your heart to lie to the Holy Spirit and keep back part of the price of the land for yourself?" (Acts 5:1-3). Since keeping back a portion of your own money intended for God is of "Satan," keeping a portion of other people's money given to God is even more so.
7. Raising funds for yourself is unbiblical.
No missionary raised funds for himself in the Bible, which instructs making requests directly to God - "Be anxious for nothing, but in everything by prayer and supplication, with thanksgiving, let your requests be made known to God" (Philippians 4:6) - who moved the Christians in Philippi to repeatedly provide for Paul during his missionary journeys: "Now you Philippians know also that in the beginning of the gospel, when I departed from Macedonia, no church shared with me concerning giving and receiving but you only. For even in Thessalonica you sent aid once and again for my necessities. Not that I seek the gift, but I seek the fruit that abounds to your account" (Philippians 4:15-17). To be Biblical, you need to ask God to provide for your needs and trust that if He wants you to stay in the field, He will move whomever He wills to provide for you.
If you are fundraising for Christians in need and can ensure that 100% of the funds raised will be used by them and none by you or your organization, you have a Biblical mandate to confidently tell rich Christians that God wants them to give all of their riches until there is "equality" between them and those Christians in need.
You lose that mandate if you ask for yourself or your organization to keep or use some of their donations. And if you keep or use some of their donations without telling them, you become worse than Ananias and Sapphira, and face an end at least as bad as theirs.