I am trying to help a player setup shop plots, however they don't seem to have access to the Party section (top right) of the image below. Also, is there a list of permission nodes available for use with Luckperms?

How can I give them access to this?

The party options are only in the "main" player config, which they probably don't have selected.
Has absolutely nothing to do with setting up shop plots.

Perhaps I'm missing something? How is this option turned on?

It's already turned on. It's not listed when you have any sub-config other than "main" selected, which I'm 95% confident is what's happening to the player you're helping.
It has absolutely no relation to setting up shop plots though.

I'm confused. each player can only create one party ever?
Wouldn't the point of sub-configs to allow for different permissions based on each plot? For example, if I was to rent a plot to another player, that would require a different party (and permission set), than say my main claim.

Sub-configs allow you to have different claim configs for different claims. That doesn't mean a separate party for each sub-config. That's not a feature and isn't ever going to be. There will be more ways to make exceptions for specific players, not just party-based, in the future but this is definitely not it.