First of all, according to traditional grammarians, it's the following that happens:
- For a single possibility, commas are used to make a non-restrictive appositive.
- For multiple possibilities, commas are not used and this makes a restrictive appositive.
The use of your particular adjective has an additional influence on the grammar of your sentence.
According to traditional grammarians (and barring a nonstandard interpretation of the use of firstborn), only the first sentence can be correct.
Ignoring the use of firstborn for the moment, people who object to the second would say that it would only make sense if they had two or more sons. When there are no commas, the appositive is used restrictively—and, in order for this to be the case, there must be multiple things from which a single item is being selected. In other words, because no commas are being used, the second sentence means that these people must have more than one son—and they are singling out Michael as the son in question.
If they have sons Michael and Jim, then the following sentence would be fine:
Their son Michael is alive.
In fact, according to traditional grammarians, the following would actually be wrong if they have two sons:
Their son, Michael, is alive.
(Because the use of the non-restrictive appositive implies that they only have one son.)
But semantically speaking (aside from a contrived example that I will follow this with), there can't be more than one firstborn son. If we assume that there is only one firstborn son, then it wouldn't be correct to use a restrictive appositive.
Here is a contrived example where, conventionally, the second sentence would be correct (and the first wrong):
The couple has two or more firstborn sons unrelated by blood. In other words, they adopted one or more firstborn sons.
Whether or not you ascribe to this traditional use of grammar is something different. Some people (correctly or not, depending on who you ask) argue that using a non-restrictive appositive doesn't necessitate a single item—and using a restrictive appositive doesn't necessitate multiple items. To those people, both of your sentences would be grammatical—and the only difference would be one of style.