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These are the sentences:
ユニオンへサンドイッチを買いに行きました。
ユニオンへサンドイッチを買って行った。
Are they both grammatically correct? Also do they both mean the same thing, which by my translation is "I went to the union and bought a sandwich."
Do they only differ in their formality with the second sentence being more casual?
ユニオン(or [生協]{せいきょう}?/大学生協?)へサンドイッチを買いに行きました。
sounds fine to me. It literally translates to "I went to the union to buy a sandwich." Its casual version would be:
ユニオンへサンドイッチを買いに行った。
Your second sentence;
ユニオンへサンドイッチを買って行った。
sounds a bit awkward to me. It sounds to me like "I bought a sandwich and went to the union." (parsed as ユニオンへ（サンドイッチを買って）行った。>> サンドイッチを買って、ユニオンへ行った might sound more natural but it also means "I bought a sandwich, and went to the union".)
買いに行った is like "went to buy" whereas 買って行った is like "bought and went".
As a side note,
ユニオンでサンドイッチを買って行った。(へ-->で)
would be "I bought a sandwich at the union and went (somewhere)"; "I bought a sandwich at the union on the way (to somewhere)"
You are comparing apples with oranges. Let me compare
サンドイッチを買ってきた。
サンドイッチを買いにきた。
instead and hopefully this answers your question.
The first construction means that one action happens after the other. You bought a sandwich and came (home).
The second construction means that you performed the second action to achieve the first. You came to buy a sandwich.
Both sentences can be stated in the polite form, without any change in content:
サンドイッチを買ってきました。
サンドイッチを買いにきました。
