Juraj:
The sensor is connected over I2C, so the API library implements register calls over I2C. You need to modify that library for Arduino (if the license allows it).
The license doesn't allow for redistribution of modified code. The nearest thing you can do is what's done here : VL53L1X on ESP-IDF - merge the API sources with code for your platform.
However, you also have to modify the API sources as they have some problems with 16 bit processors.
The API in this case is a library that provides a high-level interface to the chip functions, but relies on low-level I2C access to the device. Once you've provided that (which is somewhat similar to what Adafruit did with the VL53L0X library) then as DrAzzy says, you can call the VL53L1X API just like any other library.
I don't know of anyone except ST selling breakout boards yet. You've probably seen the earlier VL53L0X which is cheap, widely available, but not quite as good. It's pin-compatible so I hope those vendors will soon build them with 53L1X instead.
The code uses most of the code and buffer space on an older arduino. If you want to use it and have more than trivial code in your part, its probably a good excuse to try a Zero.