Is it possible to run a Photoshop action from LR?

Yes, by running a droplet.

I’ll assume you know how to make a droplet – if not, see this example – and then you can approach it a couple of ways.

One way is to save the droplet anywhere on your computer. You then set it up as an external editor on the Preferences > External Editing tab, and run it by selecting the images in LR and choosing Edit With. The droplet will be listed alongside Photoshop and any other external programs.

A second method is in the Post Processing step. First save your droplet temporarily on the desktop, then go to the bottom of the File > Export dialog where there’s a dropdown box which has an option to Show the Export Actions folder in Explorer/Finder. Choose this option and you get to the Export Actions folder, which is where you put your droplet. It will be listed as a Post Processing Step the next time you do an export.

Whether the results are sent back to LR partly depends on your droplet, because you could write it so it saves copies of files which LR might not track. But Edit With (method 1) will automatically create a file in LR, while Post Processing Steps has an Add to Catalog option.