Does God Almighty inspire His servant to supplicate, so that if a legitimate supplication comes to his mind - whether it be from tradition or not - is it God who inspired him with it?

Allah the Almighty inspires His servants to supplicate and grants them success in it and in every good, and that is among the signs of His intention to answer. So, whoever is inspired to supplicate, it is intended that his supplication be answered, based on His saying, the Most High: "Call upon Me; I will respond to you" (Quran 40:60), and His saying: "And when My servants ask you concerning Me, then [answer them], I am indeed near. I respond to the invocation of the supplicant when he calls upon Me" (Quran 2:186).