Learn Arabic (Modern Standard): سوف نطلب سمكا أو دجاجا.

No. Unlike the word for "and" (و), which always attaches directly to the beginning of the following word, أو is written as an independent, separate word. You can see it standing alone between سمكا and دجاجا.

Why do the words for fish (سمكا) and chicken (دجاجا) end with an extra alif (ا)?

Because they are the direct objects of the verb (the things being ordered). In Arabic, when a word is indefinite (meaning it doesn't have "the") and is the object of an action, it takes an "-an" ending sound called tanwin. For most words, writing this requires adding an extra alif (ا) at the very end of the word.

Could we use the prefix سـ instead of سوف to say "will"?

Yes! You could say سنطلب instead of سوف نطلب. Both mean "we will order". سوف is often used for actions slightly further in the future or to add a bit of emphasis, but they are mostly interchangeable.

How does the verb نطلب show that "we" are the ones ordering?

In the Arabic present tense, the letter at the very beginning of the verb tells you who is doing the action. The letter nun (نـ) at the start of نطلب is the standard marker that means "we".