The problem is that your "AC adapter" can't supply enough current. It is only rated for 300mA, which way to wussy. I don't know about the SB-26 stobe in particular, but other strobes I have looked at will happily draw 1 A or more after a strong flash.
Get a proper power supply rated for at least 1A. Don't play games and experiment with the voltage if you value your strobe. You can safely apply about 1.5 V times the number of AA cells the strobe is intended to take. If it's two cells, then 3 V. If 4 cells, then don't exceed 6V. 1 A may not be enough for short periods right after a flash, but is probably enough to get thru the tough charging phase a little slower to where it will draw less current after a few seconds. A little voltage sag is OK since the strobe has to be designed to at least not get hurt by low batteries. I would probably get a 2 A 6 V power supply (assuming 4 AA batteries) to get good quick recharge time.
I might (doubtful) want to try and sell this device, so I can't simply
take some Logitech web cam and use it due to obvious licensing issues.
So that's a huge constraint.
Basically you will be creating an IP Camera. The Raspberry Pi is probably the cheapest and easiest way to prototype this, however there are other boards like the Beagle Boards.
Once you get a prototype going, then you may consider creating your own all in one device that uses an ARM or DSP processor. For example I would probably use some type of Serial JPEG Camera Module, a cheap microphone, and the cheapest ARM processor that will fit these requirements. But for a prototype, the Raspberry Pi and a cheap usb web cam is probably the cheapest and quickest way to get started and get your software going. You may even to be able to find a cheap no-name usb camera from china that will work for this so you can resell it in small quantities.
So as far as getting the frames to the MCU, there's two main issues
here: (a) the choice of camera and microphone, and (b) the drivers for
connecting the cam/mic to the Pi's USB port.
Here is the huge list of devices that work with the Raspberry Pi: RPi Verified Peripherals. The USB Webcams section lists both working and problem units. Along with a bit of extra info. After you find a unit that fits your requirements (price, etc.) I would try to double check via Google to verify someone has used it and it does indeed work; although this is probably unnecessary.
I'm sure there are other units that work that haven't been tested, the two things that will help you is to make sure it is Linux compatible, and that there are ARM drivers available.
There's also the issue of A/V encoding as well as synchronizing the
video and the audio feeds together.
As I mentioned in my comment, the RP really won't have any issues handling this part. It has more than enough processing power to handle most all A/V formats.
the Pi (which would be running GNU/Debian linux) will have the right
drivers to ingest the streaming frames and send them off to a tool
that would then be able to forward them on to a WiFi or Ethernet
adapter
Basically the RP is just going to be a linux computer that is connected to the internet and has the camera and microphone plugged into it. You will install and configure libasound2-dev (for the audio) and FFMPEG to stream everything.
Then this will just be like any other server online (you may need to configure your router and port forwarding etc. to get it to be visible on the web,) and according to this, you and other computers will just access it by going to h**p://YOUR_WEBCAM_SERVER/webcam.mjpeg
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Best Answer
SparkFun has a LiPo Charger/Booster that works online.