Finding core data is not easy man and accessing them depends on the area you're working. The best place you're going to find physical copies of the cores is at oil libraries, and each one is different. For example Midland Energy Library has gone through and recorded most of the cores they have in house and have a core list downloadable in excel, which is great BUT since most core reports don't have locational info recorded on the document, MEL does not provide any locational info other than the county and field that they were able to record from the core report.
Midland Subsurface Library has cores also but they don't have any type of list of what cores they have, you can either pay them to research your request or you can visit the library and do the research yourself.
The two ways I search cored wells is identifying on scout cards if they've been cored, or using IHS to locate which wells have cores since they are the powerhouse with the best database built from the massive historical datasets purchased from PI/Dwights.
You can search their database for $25/month and download on a per well basis but just be aware that the core data you can download from them is relatively general compared to the actual detailed core reports available from the libraries.
This is just for Texas, other areas may be completely different.
Hope this helps~
IHS Scout Ticket Core Descriptions
Core Report From Library
Scout Ticket From Library
IHS US Well & Production Data
Midland Energy Library
Midland Subsurface Library