Is it permissible to deal with missionary organizations—especially those operating in social fields such as kindergartens or healthcare—with the intention of obtaining personal benefits, due to one's livelihood needs compelling them to do so?

It is not permissible to deal with missionary organizations, because their work is based on leading people from Islam to disbelief. Any Muslim who engages with them in any activity will not be safe from religious prohibitions. Therefore, one must be wary of falling into their traps.