I don't think their API as they have listed it will work for you *easily*. Their API is a 'pull' API. For what you are wanting you would really want a 'push' API, where you could configure them to call you on some notification url (or similar) when a donation occurred. This is assuming you need something 'approximating' real-time notification.
If you could have notifications delayed by up to a certain interval you could do this by having a process (you'd need to have a server where you could leave a program running for a long period) which woke up every so often, called classy to find the current list of donations (it looks like you'd need to use the activity endpoint and the look for donations, which could be a LOT of data to sift through!). Then for each new donation your process could call the twitch API to post the message.
Basically I think you could do it, but it wouldn't be pretty.