Is it possible to synchronize Yongnuo yn 560 iii flash with a Canon Speedlite 430 ex III-RT flash using in built transceivers?
The YN560 III has a radio receiver based on the Yongnuo RF603/RF605/YN560 manual flash only protocol.
The Canon 430EX III-RT has a built in radio transmitter and receiver that are based on Canon's RT protocol.
The two protocols are not compatible.
If you are using the 430EX III-RT in manual flash mode, you can also set the YN-560 III to S1 (optical slave) mode and the Yongnuo flash will fire when it detects the light from the Canon flash. If you are using the 430EX III-RT in E-TTL mode, you can set the YN-560 III to S2 (optical slave) mode and the Yongnuo flash should ignore the preflash and fire when it detects the light from the Canon's main flash.
This will let you sync the firing of both flashes without any additional gear but technically it does not use the radio transceivers of either flash since the Yongnuo is reacting to the optical output of the Canon flash. As with any time you use optical slave flashes, you may need to lengthen your shutter speed a notch or two below the camera's X-sync speed. The extra time may be needed to allow the circuitry in the optical slave flash to react to the light from the other flash.
If you are using the 430EX III-RT or any other RT "Master" on the camera's hot shoe you can attach an YNE3-RX to the shoe of the YN-560III and it will send a properly synced "fire" signal to the YN-560III. The YNE3-RX is a receiver-only trigger for the Canon RT system. When used with a Canon E-TTL capable flash (580EX II, 430EX II, YN-568EX II, etc.) it enables the full capability of the flash when using a Canon RT compatible transmitter on the camera. You'll have to make any setting changes for the YN-560III directly on the YN-560III control panel. Since the YN-560III is a manual only flash, of course you can only use it manually. No E-TTL, HSS, 2nd curtain, etc.
No. The 430EX III-RT uses a different radio triggering system than a YN-560III.
The YN-560III is slave-only in the Yongnuo 560/60x-triggering scheme. To shoot with it and the 430EXIII-RT off-camera as radio slaves, you must add an RF-603II/-605 transceiver to the foot of the 430EXIII-RT, and use a YN-560-TX, YN-560IV, YN-660, or RF-603II/-605 transceiver on the hotshoe of the camera as a master transmitter unit. And this will be manual-only triggering. No TTL, no HSS. And if you use an RF-60x as your master, you cannot remotely control the power level of the YN-560III. There is no way to manually control the power level of a remote 430EXIII-RT.
The 430EXIII-RT is a master in Canon's -RT system, but the only Yongnuo flashes that have slave capability in this system are the ones designated with -RT: the YN-600EX-RT, YN-600EX-RT II, YN-968EX-RT, and YN-686EX-RT. This system, however, will have TTL/HSS and remote power control capability.