Many non-Catholic Christians believe that we can pray to no one but God, and ask no one to fulfill an intercessory prayer request but God. But what does the Bible teach? 

In the Old Testament, Lot makes such a request of two angels, who were sent by God to destroy Sodom and Gomorrah (Gen 19:13):

Genesis 19:20 (RSV) “Behold, yonder city is near enough to flee to, and it is a little one. Let me escape there . . . and my life will be saved!”

One of them grants his “prayer” (19:21): “Behold, I grant you this favor also, that I will not overthrow the city of which you have spoken.” Someone might say, “but the angel wasn't in heaven.” Location makes no difference: the angel still fulfilled an intercessory request. 

Angels can talk to us from heaven. For example: 

Genesis 21:17-18 And God heard the voice of the lad; and the angel of God called to Hagar from heaven, and said to her, “What troubles you, Hagar? Fear not; for God has heard the voice of the lad where he is. [18] Arise, lift up the lad, and hold him fast with your hand; for I will make him a great nation.”

And we know that angels pray to God for us:

Zechariah 1:12-13 Then the angel of the LORD said, `O LORD of hosts, how long wilt thou have no mercy on Jerusalem and the cities of Judah, against which thou hast had indignation these seventy years?' [13] And the LORD answered gracious and comforting words to the angel who talked with me.

Revelation 8:3-4 And another angel came and stood at the altar with a golden censer; and he was given much incense to mingle with the prayers of all the saints upon the golden altar before the throne; [4] and the smoke of the incense rose with the prayers of the saints from the hand of the angel before God.

When Jacob wrestled with the angel (Gen 32:24-29), he asked him to bless him (32:26), and the angel did so (32:29). Moreover, when Jacob (Israel) blessed his sons Joseph and Manasseh, he not only asked God to bless them, but also an angel (“the angel who has redeemed me from all evil, bless the lads”: Gen 48:16). 

One common response to arguments like this is to note that the “angel of the Lord” is actually God Himself, or Jesus in a pre-incarnate manifestation or “theophany”: as suggested by many passages (Gen 22:10-18; 31:11-13; Ex 3:2-6, 14-16 [Moses and the burning bush]; 13:21 cf. 14:19; Jud 2:1; 6:12-14; Josh 5:14-15; Zech 12:8; Acts 8:26, 29). This is a quite respectable harmonization of the scriptural data: held by many Eastern Church fathers.

Yet in other passages this angel is distinguishedfrom God. (Zech 1:12-13 above, and the following:

2 Samuel 24:16 And when the angel stretched forth his hand toward Jerusalem to destroy it, the LORD repented of the evil, and said to the angel who was working destruction among the people, “It is enough; now stay your hand.” And the angel of the LORD was by the threshing floor of Arau'nah the Jeb'usite. 

This ambiguity (seeming equation with God in many places, and distinction from Him in others) is explained by St. Augustine, as cited in the 1911 Catholic Encyclopedia article on “Angels”:

St. Augustine (Sermo vii, de Scripturis, P.G. V) when treating of the burning bush (Exodus 3) . . . points out: “Elsewhere in the Bible when a prophet speaks it is yet said to be the Lord who speaks, not of course because the prophet is the Lord but because the Lord is in the prophet; and so in the same way when the Lord condescends to speak through the mouth of a prophet or an angel, it is the same as when he speaks by a prophet or apostle, and the angel is correctly termed an angel if we consider him himself, but equally correctly is he termed 'the Lord' because God dwells in him. . . . It is the name of the indweller, not of the temple. . . .” 

Scripture also clarifies how the angel of the Lord is a unique representative of God:

Exodus 23:20-22 “Behold,I send an angel before you, to guard you on the way and to bring you to the place which I have prepared. [21] Give heed to him and hearken to his voice, do not rebel against him, for he will not pardon your transgression; for my name is in him. [22] But if you hearken attentively to his voice and do all that I say, then I will be an enemy to your enemies and an adversary to your adversaries. [23] When my angelgoes before you, . . .” (cf. 33:14)

Angels delegated to pardon transgressions is an interesting precursor or forerunner of priestly absolution and “binding and loosing.”

Therefore, given the above Scripture and analysis we may also, I submit, plausibly interpret the “angel of the Lord” as a created angel who nevertheless profoundly “carries” God's name and presence. Such intermediaries between men and God should not surprise us since even the Law was mediated to men by angels: 

Acts 7:53 you who received the law as delivered by angels and did not keep it.

Galatians 3:19 Why then the law? It was added because of transgressions, till the offspring should come to whom the promise had been made; and it was ordained by angels through an intermediary.

Hebrews 2:2 For if the message declared by angels was valid. . . 

Angels even help distribute His grace: 

Revelation 1:4 . . . Grace to you and peace from him who is and who was and who is to come, and from the seven spirits who are before his throne...

Thus, it is a manifest fact, thoroughly established in the Bible, that angels can be prayed to and answer prayers, as a mediatorial angelic function.