Should I use "who" or "that" when referring to an organisation?

I couldn't find an authoritative source, but I found several items that touched on the question. The basic differential is that "who" refers strictly to people, and "that" can refer only to anything else:

She is the woman who raised me

not

She is the woman that raised me

Conversely,

There is the school that educated me

and not

There is the school who educated me.

The conflict arises here because you're using these companies anthropomorphically - you're happy with them, you're indebted to them. In reality, you're happy with the people AT the companies / charities, not the companies / charities themselves.

So I would go with

I am forever indebted to the charities that helped me.

or

I am forever indebted to the people who helped me

As reported from the NOAD (New Oxford American Dictionary), that is used to replace who, whom, and which; it is also used to introduce a defining or restrictive clause, especially one essential to identification.

The book that I've just written.

In the sentence I'm really happy with the insurance company that helped me with my claim, helped me with my claim identifies the company you are happy with; you are not happy with every companies. The same is true for I am forever indebted to the charities that helped me.