Yes. **“I came here by the local bus”** is grammatical, because **“the
local bus”** identifies a particular bus service or route.
Compare:
* **I came here by bus.**
→ 我搭公共汽车来这里。（一般说交通方式）
* **I came here by the local bus.**
→ 我搭当地的那班公共汽车／这里的公共汽车来这里。（特指某个 local bus service）
* **I came here on the local bus.**
→ 我搭那班当地公共汽车来这里。（更强调你实际乘坐的那班车）
So **by the local bus** is quite possible when **local** makes *the bus*
specific.
A useful rule is:
> **by + transport in general:** by bus, by train
> **by + a specified vehicle/service:** by the local bus, by the 10:30 train
The same distinction applies to **“by the train”** when *the train* is a
particular, identifiable train.