Whether or not you find Belle to be "rich" depends on your definition of the term. If you are referring to money, we don't hear much about her financial situation either way. The fact that she accuses Scrooge of idolizing money and that she breaks off their engagement as a result of his preoccupation suggests that she is not as money-driven as Scrooge.
If you are defining "rich" as "having plenty," Belle certainly has an abundant amount of love in her life. When Scrooge revisits Belle later in life, he finds her to be happy and healthy, surrounded by doting friends and family.
Perhaps Dickens was intentional in not specifying Belle's financial standing either way. It is clear that Belle values relationships over money, and that is primarily what we learn about her. We learn about Belle through the vehicle of her relationships, and we learn little about her money situation, as it is certainly not something that Belle prioritizes.