I'm at Level 20 and have finished the main story line. I am unable to play in The Crucible or participate in Strike missions because I don't have a Playstation Plus subscription. Now that the Loot Cave has been patched, is there any way for me to get legendary gear or motes of light for leveling up by completing the regular patrol missions that are still available for me to play?
It will be difficult, but yes. It is still possible. By completing bounties and patrol missions, you will eventually be able to afford legendary equipment sold by vendors, which will boost your Light level quite dramatically.
They may have patched The Legendary Loot Cave, but there are other loot caves (if you're into that). One is on the Cosmodrome. Simply spawn, turn 45 degrees to the right, and walk into the building. Go down the stairs, find a staircase, and move slightly onto the staircase. Then you just have to wait for enemies to spawn.
Dropships will mess up the spawn.