jandyman Bela has built-in classes for communicating via OSC over UDP which automatically do the processing in the background for you. The best way to handle background processing manually is to use the Auxiliary Task API (see the use of Bela_createAuxiliaryTask and Bela_scheduleAuxiliaryTask in the example here, and the documentation here).
In addition to the stereo audio input, Bela has 8 analog inputs and it is possible to use them for audio data with a few caveats. Firstly, when using all 8 inputs they are sampled at half audio rate (22.05kHz), however it is possible to use only 4 of the channels and have them sampled at full audio rate. This can be changed in the settings tab of the IDE or in the command-line arguments. Secondly, the analog inputs are DC coupled and have no anti-aliasing filters unlike the audio inputs. This can be fixed with some external circuitry, and to address this we we released an audio capelet as a Kickstarter stretch goal which is an extra cape that sits on top of the Bela cape and essentially converts the analog inputs to audio inputs. It is currently under production, and once we've fulfilled all our Kickstarter orders we are hoping to make it available.
I would say that for your application it should work fine to use all 8 analog channels to read your guitar pickups as most of the interesting parts of a guitar signal are generally below 10kHz, though you will need to build at least a level shifter to bias your signals to around +2V for the DC-coupled inputs.