Mashiach would be here. (well, at least if you mean that literally and are including all of Torah, including not speaking loshon hora and not having sinas chinam, etc.)
It’s a mitzva in the Torah to give tzedaka to aniyim, take care of orphans and widows..so yes, you will still need non-profits to properly keep the Torah. And schools and yeshivas and shuls, which are usually non-profits, will still be needed.