I see the word Senner/Sennen around a lot in the context of the Swiss. Who exactly are these people?

Senner (sing. Senn) or in French armaillis (sing. armailli, from Franco-provençal Gruyère patois armaye, meaning “cow”), are Alpine pastoral farmers, producing mostly milk and cheese (i.e. the famous Gruyère cheese) between poya (going up the Alps for summer grazing) and rindya (going back to the valley for the winter months).
Note that I have a tag dedicated to rindya (http://laurentbelkacem.tumblr.com/tagged/d%C3%A9salpe).
Even if they are no more than several hundred people in the whole country, they are still strong, traditional, popular figures in Vaud and Fribourg cantons, and that aura is very romantized by Swiss people (think Heidi with cows – an equivalent would be the Frontier Cowboy for the American soul).

Apart maybe in Jura (because they don’t have real mountains and are a bit jealous, and also are obsessed by making watches and fighting Bernese oppression) and Geneva (because outside there obsession for big black German SUVs and luxury watches, they tend to think the world revolves around their water-jet).
Culturally, the main Armaillis’ contribution to Swiss culture beside the light-blue edelweissed shirt is Father Bové’s Ranz des Vaches (https://youtu.be/zWWVMoV8UBY) which is often seen as an alternative/unofficial Swiss national hymn.

What’s the deal with the blue cotton shirts that people wear in Schwingen? Every man in my family seems to own one.

Dear anon, before all, thanks for your question.

These kind of shirt are armaillis/Senner traditional shirts, and by extension became a sort of uniform for Swiss farmers.

https://youtu.be/skpNG4myJw4

They’re available almost everywhere, from tourist traps for a huge lump of money to Landi supermarket for about CHF 25.-

On the reason all the men of your family are wearing the armailli’s shirt :
a. they’re Swiss farmers ;
b. they like Swiss traditions (and may practice competitive Swiss wrestling/ shooting/fondue eating/Hornussen/Steinstossen, among other sports) ;
c. they like traditional, no nonsense, solid garments for work and/or sport ;
d. they are leaning politically toward UDC/SVP ;
e. they misunderstood and are quite enthralled that famous italian designer Giorgio Armailli made a very Swiss collection for this winter ;
f. some or all the above.

(I’m b. & c., tag yourself) #ranzdesvaches

Hoping I have been of some help in your questioning.