All things are possible with God; however, what is important to remember is that how He chooses to do something is of paramount significance.
Yes, it is possible to come to know God and have a relationship with Him, without having first having had the advantage of having had access to the Bible. But it is rare. Here are some examples I can think of in which ones made the choice to serve God, in spite of not having had access to God's Word:
*Jethro, Moses' father-in-law. He was drawn to seek God, through knowing Moses, and examining what great accomplishments God had done through him.
*The apostle Paul.
*Nebuchadnezzar, who ate grass like a beast, until he came to realize there was a true God, and who made the choice then - to serve Him.
*The Nebuchadnezzar, who threw Shadrach, Meschech, and Abednego into the fire - came to recognize God as being the one and true God, because he saw how miraculously these 3 men had survived the fire.
*I don't think (I'm guessing) Noah used the scriptures when he preached to the people. The same in the story of Jonah - he just told the Ninevites they needed to repent.
*Many chose to serve God, after viewing the miracle of Lazarus being raised up from the dead.
*All people are able to detect the existence of God, through observing God's creation. Romans 1:20
*Samuel Morris, of some place in Africa, originally knew nothing or very little about God. But he sought God and wanted to know Him. He eventually came to know him - have a relationship with God.
*Some claim that many people in Iraq or Iraq (I forgot just where) have come to God through visions God has been sending to many of them. Maybe this is true!
But here's some verses that show that people are more likely to come to God, through the preaching of the gospel:
Rom 10:13-15
13 For whosoever shall call upon the name of the Lord shall be saved.
14 How then shall they call on him in whom they have not believed? and how shall they believe in him of whom they have not heard? and how shall they hear without a preacher?
15 And how shall they preach, except they be sent? as it is written, How beautiful are the feet of them that preach the gospel of peace, and bring glad tidings of good things!
KJV
Also, after the outpouring of God's Spirit on the day of Pentecost, about 30 days after Jesus ascended back to heaven - made it so God's Spirit was more available in the world, to help convert people, and to also more closely guide sincere Christians who serve God with their lives. In other words, God's people are led by God's Spirit in their daily lives.
Rom 8:14
14 For as many as are led by the Spirit of God, they are the sons of God.
KJV
And here's a passage that shows how the outpouring of God's Spirit in the world, helps make it more possible for unbelievers to get saved:
Acts 17:29-31
29 Forasmuch then as we are the offspring of God, we ought not to think that the Godhead is like unto gold, or silver, or stone, graven by art and man's device.
30 And the times of this ignorance God winked at; but now commandeth all men every where to repent:
31 Because he hath appointed a day, in the which he will judge the world in righteousness by that man whom he hath ordained; whereof he hath given assurance unto all men, in that he hath raised him from the dead.
KJV
Comment: God largely winked at (largely overlooked - didn't bother trying to intervene) the spiritual ignorance of people (except for the nation of Israel) in Old Testament times, due to the the fact that there wasn't the help from the Holy Spirit available to unbelievers back then, to the extent that it is now - ever since the outpouring of God's Spirit, in the early book of Acts. Now that the outpouring of the Spirit occurred, this puts places greater responsibility upon both unbelievers and believers to respond to the need to give into and accept the lordship of Christ. So that those who refuse to comply will be punished more intensely in hell, due to their knowing what's right and yet refusing it.
UNBELIEVER'S RESPONSIBILITY:
Matt 11:21-24
21 Woe unto thee, Chorazin! woe unto thee, Bethsaida! for if the mighty works, which were done in you, had been done in Tyre and Sidon, they would have repented long ago in sackcloth and ashes.
22 But I say unto you, It shall be more tolerable for Tyre and Sidon at the day of judgment, than for you.
23 And thou, Capernaum, which art exalted unto heaven, shalt be brought down to hell: for if the mighty works, which have been done in thee, had been done in Sodom, it would have remained until this day.
24 But I say unto you, That it shall be more tolerable for the land of Sodom in the day of judgment, than for thee.
KJV
BELIEVER'S RESPONSIBILITY;
See Hebrews, chapter 12.