According to page 144 of the PHB under Armor proficiency it states that
You cannot cast spells.
However the warcaster feat claims you have learned to perform Somatic components while wielding weapons or a shield in one or both hands.
Now, logic would dictate that shields wouldnt hinder Vocal Components, and you can hold a spell focus in the other hand, Why would one need to have proficiency in shields to cast, if they Have this feat?
I understand stand that RAW would dictate it is simply not possible, but I am unsure as to why.
Best Answer
Using armor or shields you are not proficient with has more drawbacks than only not being able to cast spells.
If you wear a shield without the proficiency you have...
When you pick the War Caster feat you can...
Lets look at War Caster:
The feat allows you to perform the somatic components even when both your hands are occupied. This part of the feat is focused on not requiring a free hand for your somatic components as long as at least one of your hands wields a shield or weapon.
Let's say you only use a shield and only wear it in one hand. You would be able to perform your somatic component with your free hand (no benefit from War Caster here), but in consequence of not being proficient with the shield you are not able to cast spells at all.
Conclusion:
War Caster does not allow you to cast spells when you can't, it allows you to perform somatic components without necessarily requiring a free hand. (Hands may be occupied by shields or weapons)