I would think most jurisdictions have at least the preliminary information online now.
The cities and counties I have experiences with you can pretty much look up anything online via their web site. Just basic permit records like dates, nature of work, status, value, inspection records etc...nothing too deep and it's for someone who might just want to ask "how old is this roof" or "was the front fence put in legally" type questions.
Typically I see two piles of information delineated by age. There is usually a year when they decided to automate. For example 2002, so in our city if you are looking for information after 2002, you can get basic information online from their permitting database which is just another layer of the GIS system. Anyone can look that up. If you want more information behind each permit, you have to actually walk into the building department and request them. Most times they do not question you, just an address and they can dig it up...and charge you for each page you request. It's public records and anyone has rights. Sometimes they would ask "are you the owner?" but they can't really refuse access. You could be a potential buyer, a plumber or electrician looking at old records to figure out things before repairs, a real estate agent or service for due diligence during a purchase inspection...
The records prior to the year of automation, they would have somewhere else. Sometimes it's a whole new section of the building or even a different building in a different part of the city, which may be named building archives. Those are the really old records, and many of them were never digitized and are either on paper or microfilms. If you are looking for say the original floor plans of the house in say 1057 then most likely you will find them on archives looking at microfilms.
I know one city around here they have microfilm permit information for anything prior to 1995 in archives, but from 1970 to 1976 all information is missing because termites ate the microfilms in one room. Then they have electronic information from 1996 to 2010 on their computer database but not made available online and you have to go to the building services to request those information, they will look it up on their computers and print it for you (this I believe done so they can charge you $.50 per page of whatever you need), you can also request to look at the paper records and they will bring it out for you to review, and many times you will find stuff in the paper pile that's not online. Then after 2010 you can look at online.
Sometimes you also find building information in neighbor cities or the county the city is in. I often find original building plans in the county building department because the city didn't incorporate until later. Typically the older the property the more hunting you have to do.