Hi, Dozy -
Do you have a link to the existing site?
Paypal is probably the best way to go - look at their site under support. It provides instructions how to link directly to their payment processing facility.
You'll need to set up and verify an account on behalf of the organisation.
Unless you are selling many varieties of tickets for multiple events, it's unlikely you'll need anything more sophisticated. The PayPal solution is independent of their current site although you can benefit from a dedicated page explaining what is about to happen.
You will incur charges though - so build in that extra to the ticket price.
matto (click here)