It is impossible for me to be able to demand communication power to live with my opponent 's family by joining in.(Google translation)
コミュ力 = communication power. Or in English: Communication Skill. Hence: The communication skills, that was needed (or the prerequisite communication skill) to live along with the spouse family that you married into, was something that I didn't have/possibly do/able to do/not possessed/too much for me/don't have/etc/be creative. Let me guess a NEET guy married into the female family?
How I would translate it: There's no way I could do something that demands good communication skills like marrying into a family and living with them. Here, the thing that the character is saying that he can't do (俺に出来るはずないじゃないか) is コミュ力を要求されること. 婿入りして相手の家族と一緒に暮らす is an example of something that demands good communication skills. The phrasing is not overly formal, so 'There's no way...' is an adequate phrasing I believe.