Very little is known about the details of Shakespeare's life. The traditional story is thus: No, he wasn't independently wealthy. His father was a prosperous glove maker who was prominent enough to become alderman in Stratford-upon-Avon when he was a child. However, his father seemed to fall into debt and come into some hard times later on. Shakespeare likely didn't have much from him as far as inheritance goes. Although, frankly, we just don't know.

Shakespeare was a prolific writer because he was paid to be a prolific writer. While his poetry may be a different matter, he wrote plays for audiences. Rather than sitting in wealth and idly writing on his own time.

We don't know when his career began. But he was an actor at a time when going to the theater was becoming very popular. So, acting could be a steady income if you were skilled enough. And he was, obviously, a playwright. Later, was part of his own company, the Lord Chamberlain's Men (later the King's Men), and they would split profits (and debts). Although likely not in equal shares. Having shares in your company provided a larger and more steady income. Later, they acquired their own playhouse (the Globe), which would provide an even larger income because they get to keep most ticket sales. There were also special commissions and performances. The Lord Chamberlain's Men/King's Men were invited to perform at court fairly regularly, which would be more income.

They would also make money by going on tours in the countryside. Especially when the theaters in London were closed down (as they could be in times of plague). He also profited off of "publishing" his works. A popular play might be published and he'd be paid for the rights. This wouldn't be a very big money maker though. About five to ten pounds a play.

Shakespeare mostly invested his money in property at home in Stratford.

None of this trajectory is unusual, by the way. There were a great many relatively poor people who did well by going into the theater. Neither Christopher Marlowe or Ben Jonson were from money (Johnson famously was briefly a brick layer). Richard Burbage, an actor in Shakespeare's company who also owned the theater, was the son of a joiner. And he did very well for himself and was highly acclaimed as an actor.

I have read speculation that Shakespeare's father never suffered any financial reverses. And that most of his money actually came from his father. I don't credit it very much. He didn't invest in property or attempt to buy a coat of arms (essentially purchasing the right to be considered gentry) until later in his life, which coincides with the idea that he accumulated wealth overtime.