How do you play an Eldritch Knight without armor?

[Quartz]

How about taking a level of Monk? This assumes you have a good Wis. But Fighter / Bard / EK is good if you have access to mithril armour, though I think plain Bard might be preferable.

 

[Shin Okada]

Assuming you can only use 3 core books, Mithral Chain Shirt could be a way to go. 10% spell failure chance can be while sometimes painful, within practical range.

You can also buy some Mithral Buckler. Better if you can buy an Animated one.

But frankly speaking, such a gish (warrior/mage) build is not that great without various rules (especially some feats, spells and prestige classes) and items from optional books.

 

[Dandu]

You polymorph yourself into something with lot of Strength and NA.