Well, I think "deer" and szarvas are both words for the whole family which includes many different species. It's a pretty fair word-for-word translation, and a useful pair of words to learn.
You probably have a different species come to mind as the default image when you hear szarvas in Hungary than I do when I hear deer in American, but the same thing is true of many words...
Most deer species can be called a "szarvas" (elk, moose, reindeer, caribou, etc.), with various qualifiers. But there is at least one specific, smaller species in Europe, with a much simpler pair of antlers, that is called "őz" in Hungarian. Not "the őz szarvas", no. Just "őz". It is similar to those smaller deer species that many Americans have seen several times in their lives. If you refer to an "őz" as a "szarvas", people will not know what you are talking about. They will say "Az nem szarvas, hanem őz."
So I just wanted to point out this important distinction. You can generally speak of "szarvas" and "őz" in Hungarian, most everybody knows the difference.
According to Wikipedia, an őz is a roe deer -- which also has a separate word in German, "Reh".
Interestingly enough, many people think that "Reh" is not a separate species, but the female or child of a "Hirsch" (deer in general).
Some say that Disney is partly to blame, because Bambi starred a roe deer in the original book, but for the film, they decided to use a white-tailed deer instead, as that would be more familiar to Americans.
It doesn't sound strange to me explicitly, but hunting (by itself) and "hunting for" do have slightly different implications. "Hunting for" is really just going to imply a search, not a search with intent to kill.
"I spent all afternoon hunting for this book." - makes total sense.
"I spent all afternoon hunting this book." - "Why do you want to kill that book?"
"Hunting for a book" - "egy könyvre vadászik". Yes, you can also do that in Hungarian. It means it is probably a very elusive book, hard to find, very rarely available, maybe only in certain places, you have to "literally hunt for it".
"Erre a szarvasra vadászik" actually sounds quite similar. Not necessarily an active pursuit at the moment, rather an on-going process, on several occasions. The hunter knows this "szarvas", he spotted it before, maybe he missed it a few times, it is a very elusive "szarvas", he has been stalking it for quite some time.
But it can also be just a regular "hunt".
But when the hunter spotted a deer and is tracking it, I would use a different verb, maybe "követi", "űzi", "hajtja", etc. "Vadászni" is used more as a general term for the activity of hunting.
So you're saying that "ezt a szarvast keres" means "hunt for this deer", and
"erre a szarvasra vadászik" means "hunt this deer"?
That makes some sense to me, because the word "hunt" in the first phrase really means to search for something, so "keres" fits with that. Maybe when you're tracking a deer with the hope of killing it (for food, I hope!), you have to hunt "onto" the deer. :)
That would be "keresi" (definite conjugation). But "keresi" is a very generic verb, I would not use it here. It sounds like the hunter lost his deer, doesn't know where he left it, etc. So it sounds like a "Lost deer" sign on a lamp post.
"Erre a szarvasra vadászik" can actually be pretty close to "hunts for this deer". He is trying to track down this specific, very elusive, deer.
And when he is on track, pursuing the deer, it can be "követi", "üldözi", "cserkészi", "hajtja", etc., "a szarvast".
In any case, "vadászik" is normally done with "onto sg" in Hungarian. The accusative is only used in a very specific case. You can say "levadászták ezt az erdőt", meaning they did a sweeping hunt of this forest, or, similarly, "levadászták az összes/utolsó oroszlánt", meaning they hunted down (tracked down and killed) all the / the last lion(s)".
If he is "hunting for" the deer, I would take it to mean he had it and he lost it - maybe it is in his garage or did he put it in the back shed. I would not take this to mean he is engaged in a hunt for a deer but rather searching for something he misplaced - which is not what the Hungarian means (I believe).
But, as I said above, it can mean something similar in Hungarian, too. The problem with this sentence is that we have a wild animal in it, which makes it easily associated with actual hunting. So let us change it to, say, a book:
"A férfi erre a könyvre vadászik." - means that he is trying to acquire this book, keeps going to bookstores, libraries, antique book shops, etc., it is a rare and hard book to find.
Can we say "The man is hunting for this book?"
So, at least in Hungarian, it makes a difference what the actual subject of the hunt is. If it is something that could be the target of an actual hunt, then it can be that, an actual hunt. If it is not, then it works figuratively.
One more note: I think the figurative (search) use can also take the accusative form, especially with a specific object:
"Ezt a könyvet vadászom már két éve." - I have been hunting for this book for two years already.
Yes, you can say the man is hunting for a book. What got me into this thread initially, is that they would not accept "hunt this deer" - which is natural English and the more common meaning and insisted on "hunt for this deer" - which means - like "hunting for a book" - searching not actual hunting.