I have found that the cheapest way to hold many cards is to put them in old motherboards. I have been buying motherboards off Ebay for $45, getting a $5 processor, a $100 PSU and connecting 4 cards per rig. I currently have 6 of these rigs running and they are great. As for heating I have built custom cases for my cards so they don't overheat. Using this method I've brought the cost to run each card down to about $50. In other words, it will cost about $50 per card to put it into a computer if you build your own off Ebay.
Another option, which is definitely more expensive but probably the type of thing you were hoping to find is a PCI Express Switch. These things exists and are how big companies can connect 20+ GPUs to a single computer, but they are prohibitively expensive in most cases.
Below is an example of my mining tower, if you would like more information on exactly what I buy and the economics behind it please don't hesitate to leave me a message.