CiviCRM will cover your needs and more.
As a web-based application it is available to any authorised user with a browser and a connection.
I would encourage you to read through the the section of the online user guide at http://book.civicrm.org/user/current/getting-prepared/is-civicrm-for-you/ which you may find helpful.
My only concern here is that CiviCRM - much as I love working with it - may be too powerful/feature-rich/complex for your "very small charity", and that you might end up spending more time and or money than you need to right now.
My advice would be to talk to one of the professional implementers (people like me) - here's a relevant list to get a better assessment of what you need to be able to achieve, and whether the cost benefit of CiviCRM is a good fit for your organisation.