Many small companies have difficulty growing because the number of
orders becomes too large to handle.
In your sentence, you need to see what exactly is becomes describing. Whether you should use "become" or "becomes" depends on what the word intends to describe- number or orders.
because the number of orders becomes too large to handle
Here, becomes refers to number. Number is used as singular here, so you should use becomes.
When you intend to refer to orders instead, that is when you would use become, as in this (hypothetical) example:
Many small companies have difficulty growing because their orders
become too large to handle.
Also take a look at this and this. They describe some basic rules to help deal with similar sentences.