What is the earliest level that a player character (PC) can gain the ability to become invisible without the aid of external items/scrolls/etc.?
Vanish is a spell that Wizards, Bards, Magi, and Psychics can use. Because it's a level 1 spell, all those classes can cast it at level 1 to become invisible for 1 round.
That's not for long, but Arcanists, for example, can boost this spell with Arcane Reservoir if they want to prolong it, perhaps becoming invisible for 3 rounds. If you build for it, for example, if you take the Spell Specialization and Spell Focus feats and combine it with the Potent Magic Arcanist Exploit, you can reach maximum duration right at level 1, and become invisible for 5 rounds. I don't know why one would do that, but you might be interested that it's at least possible.
Duergar can cast it as a racial spell like ability once per day from the beginning, level one:
Invisibility Spell-Like Ability (Sp): A duergar can use invisibility once per day, using its character level as its caster level and affecting itself only.
Source: Pathfinder Roleplaying Game Advanced Race Guide © 2012, Paizo Publishing
The Inquisitor Clandestine Inquisition allows you to turn invisible as a standard action lasting as many rounds as your level. You can use it 3+WIS times per day. So in theory, at 1st level, with a 16 in WIS you could use your standard action every round and move around invisible for 6 rounds.