An archivist of any level may attempt to learn Magic Jar.
An archivist scribes spells in her prayer book like a spellbook. Presuming that you have a source in the world capable of scribing from.
The specific rules referenced are here:
To learn, prepare, or cast a spell, the wizard must have an Intelligence score equal to at least 10 + the spell level. The Difficulty Class for a saving throw against a wizard’s spell is 10 + the spell level + the wizard’s Intelligence modifier.
... Therefore the Archivist must have an Intelligence of at least 15.
A wizard can also add a spell to her book whenever she encounters one on a magic scroll or in another wizard’s spellbook. No matter what the spell’s source, the wizard must first decipher the magical writing (see Arcane Magical Writings, above). Next, she must spend a day studying the spell. At the end of the day, she must make a Spellcraft check (DC 15 + spell’s level). A wizard who has specialized in a school of spells gains a +2 bonus on the Spellcraft check if the new spell is from her specialty school. She cannot, however, learn any spells from her prohibited schools. If the check succeeds, the wizard understands the spell and can copy it into her spellbook (see Writing a New Spell into a Spellbook, below). The process leaves a spellbook that was copied from unharmed, but a spell successfully copied from a magic scroll disappears from the parchment.
If the check fails, the wizard cannot understand or copy the spell. She cannot attempt to learn or copy that spell again until she gains another rank in Spellcraft. A spell that was being copied from a scroll does not vanish from the scroll.
Therefore, presuming spellcraft optimization (trivial with appropriate spells like divine guidance or a number of the bard skill enhancers from archivist) there is no inherent problem in learning magic jar.
Casting Magic Jar is really quite difficult before Archivist 7
Magic Jar is a Wizard/5. Given the house rule allowing wizard spells in the archivist's prayer book (::whimper::) there is no formal problem save for the fact that our archivist only casts 3rd level spells.
The use of the Sanctum Spell feat provides for a -1 spell level when outside of the sanctum. Optimization detailed here. This, applied to Magic Jar, turns it into a level 4 spell.
However, it is not until Archivist 7 that this becomes possible. Magic Jar is necromancy, therefore the shadowcraft mage -> miracle route when combined with divine metamagic is possible... but overkill and difficult to pull off on the budget allocated above.
While it is possible to have an artificer make scrolls using sanctum spell, the easiest way that I can see is somehow change your race to a much lower ECL and have more caster levels naturally. Given a choice between that and shadow miracles as a fake-gnome, I'd take race-fiddling.