The question is restricted to launchers that have already been used to launch an object into orbit before today (April 2018).
Some military ballistic missiles run only on solid propergol (e.g. the M51) but do not put objects into orbit. The vega mainly runs on solid propellant except for the last stage; and does put objects into orbit.
Thus my question is: do any launchers running only on solid propellant and putting objects into orbit exist?
The NASA Spaceflight article Japanese sounding rocket claims record-breaking orbital launch describes JAXA's use of its SS-520-4 sounding rocket to put the Tricom-1R cubesat (43201, 2018-016A) into orbit. While there was an issue related to communications during the ascent, the satellite is currently in a 1572 x 189 km orbit.
This is probably the smallest solid propellant rocket to put a payload into orbit. As pointed out by @PavelBernshtam answer this is certainly not the only example.
SS-520-4 is a three-stage solid-fueled rocket standing 9.54 meters tall, measuring 52 centimeters in diameter and weighing in at 2,600 Kilograms – smaller and lighter than any previous ground-based orbital launch vehicle. It is based on the SS-520 sounding rocket design, modified with a small third stage tasked with injecting a payload into Low Earth Orbit.
This is not the first attempt to reach orbit with the solid propellant SS-520-4 sounding rocket. See Will JAXA try again to launch TRICOM-1 with the “world's smallest orbital rocket” SS-520-4 again?
above left: Screenshot from N2YO.com showing Tricom-1R is indeed in orbit around the earth! Click for full size.
above right: SS-520-4 rocket ready for launch. From here, Photo: JAXA Click images for full size.
above: SS-520-4 rocket. From here, Image: JAXA
In addition to SS-520-4, I would also point to Chinese CZ-11 rocket, Israeli Shavit and many others
Adding to the existing answers, the first all-solid launch vehicle was the American Scout family. This made its first successful orbital launch on February 16th, 1961, launching Explorer 9.
