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I came across this sentence while practicing my German:

"Ich mache Ihnen deswegen keine Vorwürfe."

Which according to the language learning web site I found the sentence on, translates to:

"I am not accusing you of that."

This is confusing to me because I have never seen "deswegen" used in a sentence structure like this. My mind wants to translate this to "I make against you that is why no accusations.", which is obviously nonsense.

Does "deswegen" have another meaning or "sense" than "that is why"? Can someone tell me how to read this sentence to myself in my mind in a way that makes sense and matches the translation?

The sentence has a slightly different meaning from what you thought.

Ich mache Ihnen deswegen keine Vorwürfe.

means something like

I'm not accusing / reproaching you because of that.

The adverb "deswegen" is used in a demonstrative sense, pointing to something that was probably mentioned before.

A more natural English translation could be something like

I won't hold that against you.

Deswegen has the meaning wegen dieser Sache which can be read either as a reason

Der Supermarkt hat Sonderangebote, deswegen fahre ich dort hin

but also as "related to this issue" like in your sentence.
(for which Henning, who was a bit faster than me already posted a good english re-wording).

There is a third possibility (somewhat inbetween the others)

Ich komme 3 Minuten zu spät zur Arbeit, aber deswegen wird mein Chef mich nicht feuern

I'm 3 minutes late for work, but my boss isn't going to fire me for that.

The two answers above are correct. I want to add that, whether my suspicion about way and weg-en were correct or not, a passable translation may be

I don't blame you this way

of course this does rely on the idiomatic reading of way as "manner". That way it is less specific than "deswegen" and thus does not work with the other sense of deswegen:

The supermarket has specials on offer, thus I will go in; that's why, this' why

As far as the semantic field is concerned, compare

Fine, if you have to be this way [making such a fuzz]

Schön, wenn du deswegen gleich so einen Aufstand machen musst

Contrast

Ich verstehe nicht, warum du so dagegen protestieren musst.

Otherwise, in positive connotation, I would personally use deshalb, but I cannot realy explain why (perhaps cp "behalf", "help", Ger allenthalber, allenfalls, Gefallen, gefälligst). The contrast is not strictly observed

Ich werde deshalb / deswegen zum Sieger ernannt

Now I'd like to explain how anyway and jedweder relate through whether, ob, wieder, wider, wes- or rather was, wat, wa' ... and perhaps weigern, wiegeln, winden, eben, even, gerade deswegen ... but I am affraid this has to wait and will be more complicated than I can fathom--way complicated, in fact