Would an Exchange Server be appropriate for this situation?

Solution 1:

You should probably be looking at outsourced or hosted solutions. You can find hosted Exchange for a low monthly cost ($5 per user per month).

Exchange server will do what you want as far as the e-mail is concerned, however it's an email platform and not meant for file storage. For that you'd want to look at something like Microsoft SkyDrive, Sharepoint or even something free like Box or DropBox.

Solution 2:

If it's a charity that's using volunteer sysadmins then on-site resources and budgets are probably quite limited, I take it?

This is probably a good case for an outsourced email/storage solution such as the offerings from Google or Microsoft. For a small organisation these can allow you to implement some quite impressive tech (e.g. all the high end exchange features if you wanted) with much less up front cost and support requirements than a traditional self-managed system, which I'd imagine is important for an organisation relying on help from volunteers, where what time you have might be better spent on the more fundamental areas of support.

A solution such as Microsoft's office 365 would give you exchange (and all the shared calendar and contact list goodness that brings, solving problem #1) and the ability to store and share documents online via Skydrive Pro.