Hello, I'd like to help you with your question.
Co dependent behavior can be passed down but only as a learned behavior. It cannot be passed like a disorder such as Bipolar
both of which have genetic components to them.
Children who have a parent who is co dependent can copy the behavior but this is never a guarantee. The factors that go into whether or not a child will become co dependent are many. One factor to consider is personality. Some children are naturally geared towards being an extrovert and are more aggressive in their interactions with others (not in a bad way). They most likely would not take on the characteristics of co dependency because it directly conflicts with their own personality. Other children might take on certain characteristics while ignoring the others. And finally some children will become co dependent like their parent is as long as certain factors are there, such as the level of dysfunction or abuse that the parent suffered through (for example, parents that abuse because they themselves were abused).
It is important to consider how your wife became co dependent. Certain factors must be there in order for a child to take on something like co dependency. Dysfunctional relationships with a parent often result in a child having to adapt their own behavior in order to get their needs met. And it sounds like your wife may have had to do that with her own mother. If her mother was physically, emotionally and/or mentally abusive, this would explain how your wife ended up with the issues she deals with now. Your wife would benefit greatly from therapy to help her understand the trauma she suffered and how to see it in a another way so she can cope better.
Because your children are not exposed to their grandmother's behavior directly and they also have you in their lives to balance what they do see, it is probably safe to say that they will not develop the same adaptive behaviors your wife has had to. But if you feel there is any possibility or you are concerned, it is good to have your children see a therapist to help them understand why their mother has these issues. And it also helps for you and your wife to have open communication with your kids and let them ask any questions they want to about any dysfunctional behavior or symptoms your wife or mother in law exhibit in front of them. This will help your children avoid the same behavior patterns as their mother and grandmother and will break the cycle of dysfunctional behavior in the family.
Co dependency often results from someone who feels so low about themselves that they have to have others to "depend" on so they can feel validated. But with children, they are naturally dependent so it is very hard to separate what might be co dependency and just a child needing a parent or other caregiver. If you feel at all unsure, an evaluation by a therapist can reassure you that your children are developing normally.
I hope this has helped you,