Would answering "Jawohl" to an order or request be associated with Nazi Germany?
What about "Jawohl, mein Kommandant"? Can it be used (jokingly) without people finding it tasteless?
"Jawohl" is a normal German word, used as a strong affirmative. It doesn't have a specifically Nazi background, but one of its main uses has always been in the military, including the Wehrmacht.
drückt unbedingte Zustimmung aus (expresses unconditional agreement)
Google NGram shows it has been in use during all times since 1800.
I'd say that its use in daily conversation, however, has declined. It has a slightly outdated touch to it in many contexts. Still, this is a valid example:
Wir werden sie finden, und zwar heute noch, jawohl!
The Wiktionary link (below) and the Google Book links in the NGram have more examples of the word's use throughout the ages.
In the military, as far as I know, it is the standard affirmative answer to a superior's command in the German military to this day — it made the transition from the Wehrmacht to today's Bundeswehr. Its closest English-language equivalent in that sense is "Aye!" or "Yessir!". It often gets colloquialized into "Jawoll".
I think the idea of a Nazi connotation has a simple explanation: Most plays, novels, films, and stories that deal with the German military are set in the Third Reich. You will hear "Jawoll!" in every one of them at least once.
Also, the phrase "Jawoll, mein Führer!" is still widely used in a sarcastic way, both in the English-speaking world and in Germany (and probably everywhere else around the globe).
"Jawohl" is always considered more formal than the familiar "Ja", which is for friends and acquaintances.
"Jawohl" is the more formal version of "Ja" used very commonly in the Bundeswehr without any connotation. It is also used as shorter version for "zu Befehl" (as you order / at your command) when accepting an order, which is rarely used nowadays.
Some examples in military context can be found here. As Germany has had general conscription for quite a while, it can stil be found in civilian life.
I live in the south and I have never heard anyone say "Jawohl", except jokingly or to express uncommonly strong affirmation. It is more common in the north east, though (around Berlin).
I would never say "Jawohl, mein Kommandant" to people I don't know well, it feels like a quote from an American movie with fake Nazis.
As an American who lived in Germany (because I come from an Air Force background, but actually attended a small, local German school) I can affirm that saying "jawohl" does not carry any Nazi associations whatsoever. The word is simply a strong affirmative statement, close to when you hear what a friend is saying and say "yeah!" quickly in agreement. It is frequently used by uniformed personnel, but it is certainly not limited to them whatsoever.