USB connection without unlocking screen i am connecting my phone with USB (using MSC) to my car for listening to the music on the phone's microSD card.

It only works, when the screen was unlocked before connecting the phone.
Also, when turning off the car's engine and starting it again, I must disconnect the phone, unlock the screen, and connect it again.
This is very inconvenient.

Is it possible to allow USB connections without unlocking the screen?

Phone info: Sony Xperia Z3 with Android 5.0.2.
From comments, since you are using pin lockscreen, it wont allow you to have any type of access to internal files with usb connection without unlocking pin.

You have two choices->

1) Dont use PIN/Pattern/Password and just use simple swipe (or no lockscreen at all for that matter).

2) Disable internal lockscreen and download a third party lockscreen from play store with PIN or pattern etc support like smart app locker. As far as I can remember it supports acting as lockscreen too and not just app locker (additionally, you might also want to check this to use along with this app for extended functions). I suggest you explore this app or get some other lockscreen alternative app.

Hope that helps.