Interesting that the instruction sheet gives two bulb part numbers... Anyways, because there is no real socket, you have some flexibility. There are a whole bunch of direct fit LED options for the various LED bulbs here:
https://stores.towncountryhobbies.com/leds/
This one is specifically designed for the type of socket shown, but the socket appears to be lacking the terminals so you would have to carefully solder the leads directly: https://stores.towncountryhobbies.com/led-8352ww/
A direct fit replacement for what is shown looks to be this one and again, you would need to carefully solder it like the existing one pictured:
https://stores.towncountryhobbies.com/e5-ledww/
Hope this sets you in a direction, and don't hesitate to follow up with success or more questions!
