You definitely do not want to have multiple sites all talking to the same database. That's a recipe for a mess. Here are a few examples of things that can go wrong...
- Site 1 updates table at the same time as site 2, and there is a conflict as to what data should be saved.
- You update the code on site 1 to add a new column to the table. You do the same for site 2, but site 2's update crashes because the new column has already been added (by site 1).
- One site tries to read a record just after another has deleted it, and there is a conflict.
- etc. (there are many more cases I'm sure you can think of)
But all is not lost!!!
What you DO need to do is create a single server that is an API for the other sites. All your sites will call this single server for all updates, and that single server will be the only thing that talks to the database.
Your API will have a few basic actions - one to create a new record, one to get a record, one to update a record, and one to delete a record.
You might start by checking out the Rails resources documentation at http://guides.rubyonrails.org/routing.html#resource-routing-the-rails-default - it goes into how you can automatically generate a basic CRUD API. CRUD btw stands for Create, Read, Update and Delete - the basic functions you'll want to do.
Later, you might get more creative with your API and expand to add more functions. Suppose every time you create an order, there are several steps to be taken. What you can do is make one route in your API that you call, and the controller for that route makes all the updates necessary. Basically any time you find yourself writing code for one of your sites that makes the same set of API calls every time, you should consider creating a new route in the API that makes that call.
Another advantage of an API is that it lets you do some good thinking about what you want your database and actions to be.
You might start by first putting together the basic API, then write just one of your sites. Get it working, then add another. Consider Googling around for "Rails restful API" for some good references, and good luck!