If you are onpremises, there are several options and they do depend on the version of CRM you are running.
If you are running a recent version, you could call the CRM WebApi, see: https://msdn.microsoft.com/en-us/library/mt593051.aspx
As you are on-premises you should be able to just call the api and authenticate automatically with the current user. Maybe you need to figure out the 'internal' url of CRM to make this work without the need to authenticate. If CRM is an IFD (internet facing deployment) and ADFS is used then you don't want the 'default' url but the url that doesn't use ADFS for authentication.
To test you could just put this api in an ajax call in a Script Editor webpart and see if it works: