It's up to the DM.
The warforged race description says (E:RftLW, p. 35):
Warforged are formed from a blend of organic and inorganic materials. Root-like cords infused with alchemical fluids serve as their muscles, wrapped around a framework of steel, darkwood, or stone. Armored plates form a protective outer shell and reinforce joints. Warforged share a common facial design, with a hinged jaw and crystal eyes embedded beneath a reinforced brow ridge. Beyond these common elements of warforged design, the precise materials and build of a warforged vary based on the purpose for which it was designed.
First, to address the rules from a narrow perspective, it is up to the DM to rule on this. The description of the Shatter spell states:
A creature made of inorganic material such as stone, crystal, or metal has disadvantage on this saving throw.
This doesn't tell us how much of the creature's composition must be inorganic. Entirely inorganic? Is an inorganic prosthesis enough to give disadvantage against shatter? The rules don't answer this for us, so the DM needs to make a ruling.
I rule "no disadvantage" because warforged composition is a flavorful character design choice.
The last sentence of the paragraph quoted above is important:
Beyond these common elements of warforged design, the precise materials and build of a warforged vary based on the purpose for which it was designed.
Based on the description of the race, the percentage of inorganic composition of a warforged seems to range anywhere from 1% to 99%. A 1% inorganic warforged probably shouldn't have disadvantage on shatter, but maybe a 99% inorganic warforged should have disadvantage. But here is how I make my ruling: all flavorful design choices should be created equal, in favor of the player.
If I have an almost entirely organic warforged and an almost entirely inorganic warforged in my party, I should be treating them equitably despite their design choices, unless we have some mechanics we have agreed upon beforehand. Therefore, I rule that both warforged do not have disadvantage against shatter.
The first time this came up in a game for me, I initially thought to rule "yes, disadvantage", but we had a quick conversation mid-battle that changed my mind:
Thomas (DM): And the wizard is going to cast shatter centered on you, DEX. Con save at disadvantage.
DEX (Warforged): Disadvantage? What for?
Thomas: You're made of inorganic material.
DEX: That sucks, I could have made myself out of mostly organic material and not had disadvantage.
Thomas: Hmmm, you're right. I'm not going to punish you for making your character art look cooler with some metal stuff.