Breakdown of Вам нужны какие-нибудь новые вещи?
новый
new
вещь
thing
вам
to you (formal/plural)
нужен
needed (short form)
Questions & Answers about Вам нужны какие-нибудь новые вещи?
Why is какие-нибудь used here instead of что-нибудь?
что-нибудь means "anything" and stands alone as a noun. Because this sentence asks about a specific noun (вещи - things), we need an adjective-like word that means "any" or "any kind of". That word is какой-нибудь, and here it takes the plural form какие-нибудь to match вещи.
Could I say какие-то instead of какие-нибудь?
You could, but it changes the meaning. Words with -нибудь are perfect for questions where you don't know if the object even exists ("any things at all?"). Words with -то imply the objects exist but you just don't know exactly what they are ("some specific things"). For a general "Do you need any..." question, -нибудь sounds much more natural.
Why does the sentence start with Вам (to you) instead of Вы (you)?
To express "need" in Russian, the sentence structure is literally "[Thing] is necessary to [Person]". The person who needs something must be in the dative case. So, Вы becomes Вам ("to you").
Why is the word for "need" spelled нужны and not нужно?
The word for "need" (нужен) is actually a short adjective in Russian, so its ending must match the subject of the sentence (the thing being needed). Because вещи (things) is plural, the plural form нужны is used.