Networks

We have two systems for keeping all of our information: One (Zoho) is web- based and is more of a contact management software that can be accessed by anyone who has the log-in info. The other (Shelter Pro) is only currently available on one computer (but it can be networked, it just hasn't been). This database processes our adoptions and donations and generates receipts. Shelter Pro is the more robust of the two, but the problem is that it is available only on one computer and the person who is doing all the data input is overloaded.
Our goal is to have the functionality of Shelter Pro but have it be available via the web, without having to use a VPN or services such as gotomypc.com.
How can we accomplish this?
Thank you.