Can a rooftop a/c be added if there is not a roof vent?

Yes, this can be done and it can be done for much less than it will cost to fix the basement air (in most cases). Assuming you can get a good dedicated line to the area you would like to install the roof A/C you just need to find the supports and build a small frame on the inside to attach the roof unit and attach the interior vent. The decor on the inside might be your biggest issue but it can be done and has been done by may RVers with excellent results.