Your definition for vicinity is correct. I would suggest two things to keep in mind when using it. First, vicinity is very formal, and often used in official reports, so it may seem unusual in casual conversation. Second, vicinity is not a specific distance, but refers to a region that is understood by the speaker and listener. "All police cars in the vicinity of 45th Street and Main please respond" would be understood to be several miles, while "There were two witnesses in the vicinity of the crime" would mean close enough to see what happened.
Your definition for roundabout is not typical. A native speaker would understand roundabout to mean indirect. "I knew she would be suspicious if I asked her about her date directly, so instead I took a roundabout approach and asked if she saw last night's game. She admitted she was busy, and complained about her disappointing date."
(Alternately, a roundabout can mean a traffic circle, a type of intersection where cars flow counterclockwise rather than crossing paths.)
Your use of roundabout to mean an area almost fits the first definition - if you took a roundabout path to a destination then you would travel through much of the surrounding area, but in a very imprecise way.
It sounds like you are trying to use "around about", which is often shortened by a rural dialect to sound like "roundabout". The phrase "around about" indicates something that is somewhat close, with a significant margin for error. Consider the following:
When will you pick me up?
I will pick you up at 7:30.
This means exactly 7:30, with no margin for error. The speaker is either very certain that they can arrive, or believes that they will be forgiven for being off by a few minutes.
I will pick you up about 7:30.
Here the speaker has explicitly said that they will arrive near to 7:30, rather than at 7:30 "on the dot".
I'll be there ... hmmm ...around about 7:30.
Here the speaker has taken the already imprecise "about 7:30" and added an additional word meaning the estimate is not precise. This phrasing is kind of rural, and indicates someone who is very casual and "laid-back".
If we take that usage and apply it to a region, as in your example, we can understand how someone might use it to mean "vicinity".
There's an awful lot of deer around about that lake.
In this case, the speaker is communicating that the deer aren't at the lake, but close enough for the lake to be a landmark. If someone were to head for the lake and wander around a bit, he would expect to see some deer.