No, a USB Monitor does not require a VGA card.
I have several small intel x86 machines with no display adapter (VGA Card) of any kind. However, a large quanity of the ones I own come with an additional module in the BIOS firmware that will redirect the BIOS SETUP to the COM1/dev/ttyS0 serial port.
Even this is not required -- I have some jetway motherboards that have a 'headless' option that can be turned on in the BIOS. However, there is a catch 22 -- you must plug in a display adapter to configure the setting in the BIOS, then remove the display adapter again afterwards.
Many small form factor (Mini-ITX) motherboards will have similar options, they tend to be popular with home NAS builders for networked storage.
The BIOS will never see most USB Framebuffer devices as a viable display adapter (In many cases, the firmware ROM chip is not included with cheap USB framebuffer devices as a cost-saving measure, relying on a firmware to be sent over USB to the device's memory to start up) however, if you can get board to boot linux in some fashion, the USB monitor should light up once it's discovered, probed, and had firmware thrown at it from the drivers.
I have a ViewSpan UFX7000 based USB 3.0 adapter that I commonly use to plug a monitor into headless units. On modern Ubuntu 13.10 and 14.04, it's detected as a new display device as soon as I plug it in and the Login Greeter displays, and then I plug in a USB keyboard and mouse and they're correctly routed to that display, and I log in to the desktop environment.
I've also tested a couple DisplayLink based devices, they behave the same when displaying the Login Greeter but the older USB 2.0 displaylink devices that lacked full compression support are often too choppy to be usable. (DL-160 was choppy, DL-195 was okayish.)
However, the newer displaylink DL-3000 and DL-4000 series are not supported under linux when last I checked due to some kind of encryption, compression, or encoding process that was not understood by the public developers and unwilling to be released by the hardware manufacturer.
The SMSC ViewSpan UFX6000/7000 is partially supported under linux, however the company SMSC was purchased by Microchip sometime last year. They tried to release some updated drivers in February 2014, but someone botched the job and the updated framebuffer drivers are not available due to a File Not Found error. http://www.microchip.com/wwwproducts/Devices.aspx?product=UFX7000 http://www.microchip.com/SWLibraryWeb/product.aspx?product=SRC-UFXx000_FBLINUX
I sent a mail to webcorrections >AT< microchip.com asking for it to be updated back in march, but there's been no change as of May 2014.
Edit: It's August 2015, I sent a mail to webcorrections >AT< microchip.com asking for it to be updatedchecked back in march, but there's been no change as of May 2014and the framebuffer driver package is downloadable now.
The xorg GUI drivers apparently have not been updated by the company since 2011. http://www.microchip.com/SWLibraryWeb/product.aspx?product=OBJ-UFXx000-XLINUX
But if you can get the framebuffer itself working, loading the fbcon module will work well enough to get a text login screen.
It's probably a good idea to have a display adapter on hand for debugging, but once you get it all set up, it should no longer be required and can be shelved or returned to it's source machine, if you borrowed it for a bit.
Your mileage may vary!