The people who have never heard the Gospel of Christ most certainly can and will attain eternity with God. This is clarified in a few places, but the most common place to read scripture is Romans 2:12-16

Romans 2:12-16 = "For all who have sinned without the law will also perish without the law, and all who have sinned under the law will be judged by the law. For it is not the hearers of the law who are righteous before God, but the doers of the law who will be justified. For when Gentiles, who do not have the law, by nature do what the law requires, they are a law to themselves, even though they do not have the law. They show that the work of the law is written on their hearts, while their conscience also bears witness, and their conflicting thoughts accuse or even excuse them on that day when, according to my gospel, God judges the secrets of men by Christ Jesus."

God judges man's heart which is the motivational source behind any person's actions. So those who don't have "The Law" (meaning, they've never heard the Mosaic Law found in the first five books in the OT, or heard the Gospel of Jesus Christ), still have "written on their hearts" the natural law, or can be better described as "common sense". For example, you don't have to hear or know the story of Christ to feel compelled to help someone lying on the ground helpless with visible injury. Nor do you need to have read the Ten Commandments to know it's wrong to murder another human being. This things are "common sense" or "written on our hearts" from God, in which God uses on the day of judgement.

Even Jesus himself showed this example every time it's recorded that he interacted with certain individuals. He's been tested on several occasions and he always penetrated through the "doing" of that individual, and went right to the "source" of the individual's doing - the heart. He never condemned the action which the person was guilty of, he condemned the hardened heart. I believe this is further confirmation that people who've never heard of Jesus will still get a fair judgement from God, and included in judgement is always God's unimaginable passion, love and unwavering grace.

I'll provide answers from a couple charismatic theologians, Wayne Grudem and J. Rodman Williams. Williams is the author of the first charismatic systematic theology, Renewal Theology, and I'll quote from it and some of his other writings.

Original Sin

First, both writers affirm the doctrine of original sin. Williams describes it as follows:

"Original sin" refers to the fact that the human race is sinful in nature. [...] [T]he important feature is that man does not come into the world as an innocent or neutral creature but is affected by sin in all aspects of his being (Psalm 58:3; cf. Psalm 51:5). Indeed, by virtue of this fact, man is vitiated in every area of his nature—body, soul, spirit—so that he is utterly incapable himself of restoration and salvation. His only hope is in Jesus Christ.¹

Referring to the people who are "without excuse" from Romans 1, Williams writes:

These are people of all times and places, who on their own account and to their own guilt and judgment, turn away from the living God.²

The clear implication here and in the rest of Williams' writing is that the unreached face judgment.

Grudem agrees, and writes more regarding the general revelation mentioned in Romans 1:

[E]ven without the Bible, all persons who have ever lived have had evidence in creation that God exists, that he is the Creator and they are creatures, and have also had some evidence of his character. As a result, they themselves have known something about God from this evidence (even though this is never said to be a knowledge that is able to bring them to salvation).³

Salvation of Infants

However, both Williams and Grudem specifically refer to the case of infants as a potential exception to this rule. First, Williams writes:

I suggest you look at Matthew 18:2-4 and 19:13-14. Note especially the words "Let the children alone, and do not hinder them from coming to Me; for the kingdom of heaven belongs to such as these." Since Jesus graciously received them during His ministry, surely He will do so in heaven. This does not mean that little children, or infants, are innocent, but they have no ability to make a decision for or against Christ, which is the requirement for salvation.⁴

Grudem goes into more detail. He says that even infants have sinful natures, citing Psalm 51:5, but he notes the example of John the Baptist in Luke 1:15, saying:

Yet it is certainly possible for God to bring regeneration (that is, new spiritual life) to an infant even before he or she is born. This was true of John the Baptist [...] We might say that John the Baptist was "born again" before he was born! [...] It is clear, therefore, that God is able to save infants in an unusual way, apart from their hearing and understanding the gospel, by bringing regeneration to them very early, sometimes even before birth.⁵

He cautions against taking this too far, but finds this as evidence that infants who die may be saved:

We must, however, affirm very clearly that this is not the usual way for God to save people. Salvation usually occurs when someone hears and understands the gospel and then places trust in Christ. But in unusual cases like John the Baptist, God brought salvation before this understanding. And this leads us to conclude that it certainly is possible that God would also do this where he knows the infant will die before hearing the gospel.⁵

Summary

In summary, these two charismatic theologians believe in original sin, and thus that judgment is due to all men because all men have sinful natures and act sinfully. However, they suggest that provision may be made for children who die in infancy. One could speculate, based on this reasoning, that similar provision might extend to the mentally disabled, but neither author addresses that situation. But extending this logic to mean salvation for the unreached peoples of the world is certainly outside the teaching of these two theologians.

References:

1. Williams, Theology Q and A, page 31 (See also Renewal Theology, 1: chapter 11, "The Effects of Sin.")
2. Williams, Renewal Theology, I-272.
3. Grudem, Systematic Theology, ch. 7, p122.
4. Williams, Theology Q and A, page 32
5. Grudem, Systematic Theology, ch. 24, p500.