Learn Icelandic: Við borðum hér eða við förum heim.

No. The word eða (or) is a coordinating conjunction, which means it just links two normal sentences together. Both halves of this sentence keep the standard subject-verb order: Við borðum and við förum.

Could I drop the second við in this sentence?

Yes! Just like in English, if the subject is the same in both halves of the sentence, you can leave it out the second time. Saying Við borðum hér eða förum heim is completely natural.

Why is the word for home heim instead of heima here?

Icelandic uses different words for location versus movement. You use heim when you are moving towards your home, which pairs perfectly with the verb fara (to go). If you were already resting at home, you would use heima.

Why do the verbs borðum and förum both end in -um?

In Icelandic, verbs change their endings based on who is doing the action. The -um ending is the standard rule for the pronoun við (we) in the present tense. Whenever you see við, the action word right next to it will almost always end in -um.