For anyone who stumbles here on a Mac, here is the config that works for me with the Uno:

{
    "env": {
        "arduino_path": "/Applications/Arduino.app/Contents/Java",
        "arduino_avr_include_path": "${env:arduino_path}/hardware/arduino/avr",
        "arduino_avr_include2_path": "${env:arduino_path}/hardware/tools/avr/avr/include",
        "arduino_avr_compiler_path": "${env:arduino_path}/hardware/tools/avr/bin/avr-g++"
    },
    "configurations": [
        {
            "name": "Mac",
            "defines": [
                "ARDUINO=10810",
                "__AVR_ATmega328P__",
                "UBRRH"
            ],
            "includePath": [
                "${workspaceRoot}",
                "${env:arduino_avr_include_path}/**",
                "${env:arduino_avr_include2_path}/**"
            ],
            "forcedInclude": [
                "/Applications/Arduino.app/Contents/Java/hardware/arduino/avr/cores/arduino/Arduino.h"
            ],
            "intelliSenseMode": "gcc-x64",
            "cStandard": "c11",
            "cppStandard": "c++11",
            "compilerPath": "${env:arduino_compiler_path} -std=gnu++11 -mmcu=atmega328p"
        }
    ],
    "version": 4
}

The key to finding <avr/pgmspace.h> was the adding the hardware/tools/avr/avr/include path.

Defining ARDUINO=10810 was identified from the output of running Arduino: Verify with the verbose flag.

Defining __AVR_ATmega328P__ was added to allow proper IntelliSense completion of the raw register macros (e.g. _BV(), OCR0A, TIMSK0, etc.); the correct macro to define was identified based on the chip being an ATMEGA328P-PU and by inspection of the file hardware/tools/avr/avr/include/avr/io.h.