I want you to increase my salary because I cannot look after my family. Your statement is written collectively, a demand on behalf of a number of people, regarding a number of families that need to be 'kept' It is also incomplete. 'Keeping' your family is only an outcome if the alternative is 'losing' your family. You are saying 'If my salary does not increase, my family will be gone' You can use 'keep' but it needs to have some other subject. 'we cannot keep our families safe' or 'we cannot keep our families from going hungry' or 'we cannot keep our families warm'

We ask that you increase our salaries so that we can adequately support our families.

Hm, there's nothing grammatically incorrect, however it doesn't really make sense. I think what you're trying to say is something to the effect of "because we cannot keep our families fed"?