I think this is a simple question, but my text book failed to clarify it.

Let us say you have a sentence that involves using に with two preceeding objects, e.g. 「妹も姉さんに絵葉書を出した。」

Another form my text book has used is, 「妹にも姉さんにも絵葉書を出した。」

My main question is, what is the difference between a final に, and placing に after each noun before the joining particle? Is this simply a question of style?

A guess would be that the former case indicates one action directed at both, and the latter case indicates an action directed at each.
In this case, that would mean one postcard vs. two.

Secondly, I have not so far seen an example of '<a>にと<b>に(と)[...]'
Can と be used in the second manner also? What about や?

>「妹も姉さんに絵はがきを出した。」

In this sentence, it is 妹 who sent a postcard to 姉さん, and the use of も implies 妹 is not the only person who did so.

>「妹にも姉さんにも絵葉書を出した。」

Here, someone sent a postcard to each of 妹 and 姉さん.

>Secondly, I have not so far seen an example of '<a>にと<b>に(と)[...]'
>Can と be used in the second manner also? What about や?

Unlike も, と and や are only used to combine two nouns, so you can't place と after '<a>に', for example. Instead, you can say '<a>と<b>(と)に' or '<a>や<b>に'.

> In this sentence, it is 妹 who sent a postcard to 姉さん

Oh, of course. Thanks.

So would "妹も姉さんもに[...]" be the case of sending a single postcard addressed to both of them?

No, that's not a valid sentence. In general, も introduces a topic to the sentence, just like は does. Consequently, も is a sentence-level structure, so '妹も姉さんも' does not form a noun clause, which に demands.

If you want to say that you sent a single postcard to both of them, you can say '妹と姉さんに絵はがきを出した'. However, this is still ambiguous because it can also mean sending one postcard for each of them. If you want to be explicit, '妹と姉さんに宛てて絵はがきを出した' would be a choice.

To whom did you send a postcard ?
だれにはがきを出したの？
　
妹と姉とに出した。
妹と姉にと出した。
you can use both of them.

×妹と姉にと出した。
△妹と姉とに出した。
○妹と姉に出した。