Is it possible to set it to where, if someone was to pay for a rank on a custom made website, that it could send the info over to LuckPerms on the server, to basically "autorank" said player?
e.g.

If i setup a purchase-able rank on BuyCraft or Enjin or something. And someone was to buy said rank.
then Enjin or Buycraft would automatically be able to send a command to the server like /lp user promote

You can already do promote. The wiki page for that is called “Tracks”.

Otherwise /lp user <user> parent set <group> or /lp user <user> parent add <group> are an option too

@BrainStone But how would the website know which ingame player had bought the rank?

Any Minecraft-Shop system requires you to enter your username. And that information is sent to the server.