Is there any way to access a local server via a smartphone during developement, or must I test it online?
Yes, of course. If your smartphone can connect to your local network over wifi then it can connect to any device on that network (that accepts connections), including your server, just like any other computer can. I do this to keep tabs on mobile compatibility during development, but it is limited to the phones(s)/tablet(s) in your possession so it isn't a very thorough way to test.
If you are using a Mac, you can give Sauce a try.
All the browsers. Right here. Right now.
Sauce will provide you a virtual machine to test on every platform a local/online website, which every browser. The free version is restricted to few platform, but it might be enough for you.
Otherwise, you can configure access to your local server over your wifi. Just type the IP address in the URL bar.
If you just want to see how it renders in a responsive way, you can test it on the last Firefox release.