I have a feeling that there is not a Christian denomination on the face of the earth that does not believe God can do absolutely anything he wants to do-- anything, that is, within the scope of his character and attributes.
It is a trustworthy statement:
For if we died with Him, we will also live with Him;
If we endure, we will also reign with Him;
If we deny Him, He also will deny us;
If we are faithless, He remains faithful, for He cannot deny
Himself (2 Timothy 2:13 NASB, my emphasis).
Yes, there are many things which God cannot do. He cannot lie. He cannot go back on a promise. He cannot sin. In short, God cannot not be God.
Having said this, there is, then, no great leap required to conclude that God can choose to appear to his creatures as a flesh-and-blood human being, as he apparently did to Abraham (and Jacob, and Moses, and others), or as a pillar of smoke or fire, a burning bush, the angel of the LORD, a mighty and rushing wind, a dove, a voice from the sky, or a little baby boy who is conceived by the Holy Spirit and born of a virgin.
To invoke Trinitarian concerns into the mix is to muddy things up unnecessarily. Some theologians talk about "Christophonies," or pre-incarnate appearances of Jesus Christ. Perhaps they are onto something. I don't know. What I do know is that God can do anything he wants, simply because he is God.
Too often when thinking about the omnipotence of God some people think perhaps of a spectacular, miraculous, impossible-for-any-human-to-accomplish feat of raw strength and naked power. And, indeed, Jesus did make the wind and the waves obey him (Luke 8:22-25; Mark 4:35-41).
One logical conclusion of that sort of reasoning would seem to be this, however: If God is able to control and manipulate the awesome forces of nature however and whenever he wills, why should even the worst doubter's natural incredulity be further stretched to believe that God could appear as a man and eat a meal? (Argumentum a minore ad maius.)
Or as King Solomon realized millennia ago,
But will God really dwell on earth? The heavens, even the highest heaven, cannot contain you. How much less this temple I have built! (1 Kings 8:27)
When God chooses to appear as a man to Abraham, is that man truly God? Of course not. Just as a temple cannot contain God, neither can a human body contain God. Even the Lord Jesus Christ, although he appeared to us as a flesh-and-blood human being who experienced everything we experience as human beings--except for sin (Hebrews 4:15) emptied himself, that self-emptying was only for a time, and for the suffering of death (Philippians 2:6-8). After his resurrection, however, his Father restored him to the highest place in heaven (Philippians 2:9-11), which is befitting only to the Son of God!
Again, Argumentum a minore ad maius.