Is it possible to transfer data from my old Mac's hard drive to my new Mac if I no longer have my old Mac?

If you do not have a backup (time machine or other external drive) or the actual old hard drive, no, there is no way to access the info on the old one. It may have already been wiped. You need to have access to it or its contents.

Although the instructions detail how it is done Mac-to-Mac, you can indeed use a 'regular' locally attached hard drive (NOT just a Time machine backup) as the source of the information to be transferred.

When you go to specify the source, local drives are typically shown as well. Just select that drive (not calling it a time machine drive).

Babowa is correct -- without physical possession of that drive, there is no way to access the old data.

My answer above presumed that you Do indeed have the old drive in your possession, removed from the old Mac and installed in an external enclosure, and physically connected to the new Mac.

if you turned in the drive with your only copy of your files when you turned in your computer, your data are GONE.