Just skimming over all the abilities and their morphs, I spotted a resounding two sorc abilities that cost stamina: Bound Armaments (morph of Bound Armor) and Dark Exchange/Dark Conversion. The second one doesn't really count though because it exchanges stamina for other resources.
So again: Possible? Yes. Yes, if you don't care to use any if your sorcerer skills. Viable? Questionable. Painful? Probably.
A lot of them are probably floating around in the mystical realm of 'Delete Character'. Or maybe through the power of defying all logic, through the act of spending attribute points in stamina as a Sorcerer class, they acquired the ability to become invisible and are working with Sheogorath as his disciples of madness to devise a plot to flood Tamriel with unseen chaos. Armed with Wabbajacks and seated atop Unicorn mounts, they wait patiently to strike and turn us all into cheese...
Stamina Sorc is perfectly fine.... just use their magic for buffs and utility such as Thundering Presence, Power Surge, etc., and pick up Stamina based weapon skills. Done. It's actually not bad, yet not a top performing build.
Stamina sorcerers used to be good and fun... now they are mostly fun. The 1.7 buff system essentially broke them (no way to stack weapon power buffs, no reliable healing with critical surge needs, changed passive expert mage to be of zero use to stamina builds, and the uncapping of stats. All making magicka stacking by far the optimal choice for sorcerers).
I currently run one as a tank or DPS depending on situation, because I dislike the magicka build style. It is still fun, but you are somewhat behind the optimal curve - still capable of soloing craglorn with 90cps tho... well most of craglorn anyway.
The main issue they have is lack of synergy with class passives and lack of class morphs. So they have no stamina scaled damaging or defensive abilities. There are 2 utility stamina morphs, one increases Max stamina as a toggle (2 slots), and the other is a channelled heal / resource swap (which no-one uses because it essentially snares and silences you).