Learn Polish: Czekamy na dostawy warzyw. - We are waiting for deliveries of vegetables.

Because it is in the genitive plural, representing "of vegetables". The dictionary form is the neuter noun warzywo. To form the genitive plural of most neuter and feminine nouns, you drop the final vowel, leaving what is called a "zero ending". So, warzywo becomes warzyw.

The lesson mentions a "mobile e" (like matka becoming matek). Why doesn't warzyw get an extra "e"?

When you drop the final vowel to make a genitive plural, you are sometimes left with an awkward group of consonants at the end of the word. Polish inserts an "e" to make it easier to pronounce (like the "tk" in matka becoming matek). When the -o drops from warzywo, the end of the word is just -w following a vowel (-y-). This doesn't create a hard-to-pronounce consonant cluster, so no extra "e" is needed.

I remember learning that waiting for something uses the genitive case. Why are we using the accusative na dostawy?

The verb czekać (to wait) on its own takes the genitive, but that usage is mostly old-fashioned. In everyday modern Polish, you almost always use czekać with the preposition na, which requires the following noun to be in the accusative case.

If dostawy is in the accusative plural, what would this sentence look like if we were waiting for just one delivery?