I heard that wealthy people won't make it to heaven because their wealth is their god. Is this true? Do wealthy people have a chance for salvation?
The answer is yes, it is possible for a rich man to be saved by grace, just as all men may be saved. The Scripture includes numerous accounts of men who were wealthy and yet who also followed the Lord faithfully.
Perhaps the best example of a believer who was wealthy is Abraham (e.g., Genesis 12). The Bible testifies he had many sheep, oxen, donkeys, camels, and servants, and yet he was called the father of faith.
Wealth in and of itself is not evil. The Bible says the love of money is the problem. In 1 Timothy 6:10, Paul says this:
For the love of money is a root of all sorts of evil, and some by longing for it have wandered away from the faith and pierced themselves with many griefs.
As you say, for some wealthy people, their wealth is their god. Jesus tells us in Matthew 6:24:
No one can serve two masters; for either he will hate the one and love the other, or he will be devoted to one and despise the other. You cannot serve God and wealth.
A person who makes wealth his god clearly is not devoted to the true God. Probably the best example of this is when Jesus gives the gospel to the rich young ruler in Matthew 19:16-26:
And someone came to Him and said, "Teacher, what good thing shall I do that I may obtain eternal life?" And He said to him, "Why are you asking Me about what is good? There is only One who is good; but if you wish to enter into life, keep the commandments." Then he said to Him, "Which ones?" And Jesus said, "YOU SHALL NOT COMMIT MURDER; YOU SHALL NOT COMMIT ADULTERY; YOU SHALL NOT STEAL; YOU SHALL NOT BEAR FALSE WITNESS; HONOR YOUR FATHER AND MOTHER; and YOU SHALL LOVE YOUR NEIGHBOR AS YOURSELF." The young man said to Him, "All these things I have kept; what am I still lacking?"
Jesus said to him, "If you wish to be complete, go and sell your possessions and give to the poor, and you will have treasure in heaven; and come, follow Me." But when the young man heard this statement, he went away grieving; for he was one who owned much property. And Jesus said to His disciples, "Truly I say to you, it is hard for a rich man to enter the kingdom of heaven. "Again I say to you, it is easier for a camel to go through the eye of a needle, than for a rich man to enter the kingdom of God." When the disciples heard this, they were very astonished and said, "Then who can be saved?" And looking at them Jesus said to them, "With people this is impossible, but with God all things are possible."
While it may be difficult for one who is devoted to their wealth to set it aside, if necessary, and come to faith in Jesus Christ, the same barrier exists for all men in one way or another. This is why the Apostles asked the question above and why Jesus responded by saying that the impossible is possible for God. The truth is no one can be saved in his own power to live righteously and choose God over our earthly desires. Salvation comes by faith alone, and faith is the free gift of God (Ephesians 2:8). That is why Jesus says it is impossible for men, and possible only with God.
Note that the reverse of Matthew 6:24 is also true. If a person loves and is devoted to God, he will despise wealth. That is to say, he will not put his confidence in his wealth or rest in it. We see this with Abraham as well, as Scripture tells us he never saw this earth as his home. Hebrews 11:9-10 puts it this way:
By faith he lived as an alien in the land of promise, as in a foreign land, dwelling in tents with Isaac and Jacob, fellow heirs of the same promise; for he was looking for the city which has foundations, whose architect and builder is God.
Abraham was never devoted to his wealth. He always looked forward to heaven, where his true home was.
So, in summary, Scripture does not tell us that a wealthy person cannot go to heaven. Rather, God says that a person who devotes himself to wealth and to gaining more of it cannot be a servant of God, because such a person is giving evidence that money is their master, and he cannot serve two masters. People who serve wealth (or make it their god) clearly do not seek to serve God because they do they know Him truly. In all likelihood, these people will not go to heaven, unless and until they come to know the one true God in the Person of the Lord Jesus Christ.