Is it orthodox to abandon the concept that God of the Bible is person, but say that He is spirit?

Here is an answer directed to you personally in regard to your question. God the Father is omniscient (all-knowing). The Holy Spirit is omnipresent (all-present). The Son is omnipotent (all-powerful). Because they are a trinity any of these attributes can be identified with each of the persons of the trinity.

As has already been stated personhood does not necessitate a physical form. Jesus is credited with creation well before He took on flesh.

While this answer will not resonate well with this site one can understand that Jesus Christ’s identification of God as Father and Himself as the only begotten Son does identify God as Family. God is not a person; God is however three persons, yet one God.

Just as there are family dynamics that we naturally understand, within the Godhead these same family dynamics exist.

So God the Father was never a man, however Jesus Christ only does what He sees the Father do. In other words Jesus is the image of the invisible God, He is the manifest will of God the Father. The Holy Spirit proceeds from God the Father bearing His will. The Holy Spirit brings to fruition the will of God the Father. Jesus Christ is that manifest will born of the Holy Spirit.

Jesus never once mentioned the Holy Trinity, Jesus repeatedly mentioned the Kingdom of God, His Heavenly Father and His own son-ship.

So the three persons of the Trinity are God, God the Family.

There's too much that we do not know about the LORD and would not even understand with our finite minds. The little I have come to understand is the following:

God is one. (Deut 6:4)

Hear, O Israel: The Lord our God, the Lord is one.

The Hebrew word for 'one' here is the same one used in this passage.(Genesis 2:24)

 That is why a man leaves his father and mother and is united to his wife, and they
become one flesh.

When the Bible speaks of the LORD's oneness, it is speaking about how He is united, of one accord, one mindset, one purpose. When two people are joined in marriage, they become one flesh. This does not mean that they have one body and are one 'person'. This means they are now (or at least should be) of the same mindset, same accord, same purpose.

Now, God is not 'one' meaning the same body. God is 3 co-existent (have always existed) People as in 3 Spiritual Beings, One of whom willingly took human shape to save the fallen humanity. (Philippians 2:6,7)

Who, being in very nature God,
    did not consider equality with God something to be used to his own advantage;
rather, he made himself nothing
    by taking the very nature of a servant,
    being made in human likeness.

He is a spirit, that is, not flesh and bone like us. Yet we know a lot about His character because Jesus came to show us what the Father was like through Himself. This is why when Philip asked Jesus to show them the Father, Jesus answered: (John 14:9)

    Jesus said to him, “Have I been with you so long, and yet you have not known Me,
Philip? He who has seen Me has seen the Father; so how can you say, ‘Show us the Father’?

There is a lot that we don't understand about the nature of God the Father and probably never will. The most important thing is to focus on what has been shown to us through the Bible. That Jesus, being God, made Himself a man in order to die and and pay for the sins of humanity so that you may have a chance to live with Him and reign with Him. The question is, have you accepted Him as your Savior and King? He has done it all, but you still must choose.

In a mainstream evangelical understanding, God's essence does not consist of anything created. Matter is created. God is spirit and existed as such before the creation of matter or time or space. A "body" is something physical, and only the Second Person of the Trinity has a body, which He received at the Incarnation.

Personhood does not mandate a physical body. Every creation is lesser than its creator. Thus, we could not have personhood unless God first possessed Personhood.

So, in summary, God is spirit, which is not matter, and He does exist in Three Persons, only One of which has a body, but He did not always have one, receiving His body at the Incarnation, which was later glorified at His resurrection.