A "medium", as a noun referring to a person, means "someone who claims to be able to contact the dead, via some special psychic ability". So, don't use that.
The words you listed (mediator, intermediary, agent) can all apply to organisations as well as individuals, "mediator" & "intermediary" more so than "agent" (the company would typically be called an agency instead).
Another word used to describe a company which mediates between two parties is a "factor", though this may be purely a British usage.
If you could provide a link to the website of an example of this type of company that will help.
I just remembered that "Mediator", as a description of a company, tends to be used (I think) more in situations where there is some hostility between the two other parties - such as a difficult divorce, for example. This reinforces the choice of "Intermediary", i think, in a situation where there the two sides are trying to help one another, and need some help doing it. You could also use "Facilitator", which suggests that the "inbetween" company has specialist skills which are needed to complete the negotiation (which, again, may be the case for you). To spell this out a bit more:
"Intermediary" - acts as a shared point of contact between two parties
"Mediator" - acts as a shared point of contact between two parties which have some hostility towards one another (and thus need to be kept apart to avoid arguments)
"Facilitator" - acts as a shared point of contact between two parties who want to achieve the same thing together, but need some expert help to do so.