the "source" is more evident if you punch the coordinates (32°40′47″N 101°43′51″W) into topo software such as www.caltopo.com. It's the drainage just to the SE of that road intersection. However, you'd have to go far, far downstream to find the head of canoeability. There's a gauge just upstream of Lake JB Thomas (USGS# 08117995), at FM Road 1205. That looks like it typically carries about half a foot except when it storms; so somewhere around there is about as far upstream as you could possibly put in. But, with a series of reservoirs downstream, you could be walking some places if they're not releasing much water.
Start here and you can explore the various gauges to get an idea of how much it flows:
Some of the pictures available online of the upper sections look pretty bleak from a paddler's perspective. But I'm sure it would be an adventure to explore.