Learn Croatian: Eto tvoje narudžbe!

It depends on who the object is closer to. Evo is used for things near the speaker ("Here is..."), while eto is used for things closer to the listener ("There is..."). Since the speaker is pointing out your order, which is likely sitting near you, eto is the perfect fit.

Why is the phrase tvoje narudžbe in the genitive case rather than the nominative tvoja narudžba?

The presentational particles evo, eto, and eno always require the genitive case for the thing being pointed at. Because narudžba (order) is a feminine noun, both the possessive pronoun tvoja and the noun itself change their ending from -a to -e in the genitive singular.

Where is the verb in this sentence?

There isn't one, and it doesn't need one! Presentational particles carry the meaning of "here is" or "there is" all by themselves. Adding a verb like je (is) would be grammatically incorrect here.

If a waiter is physically handing me my order, would they still say Eto tvoje narudžbe!?

Probably not. If the order is currently in the waiter's hands, it is closest to the speaker, so they would naturally say Evo tvoje narudžbe!. They would use eto if the plate was already placed down on your side of the table, or if they were pointing to it sitting on the counter next to you.