I have not heard his name mentioned in the Veep stakes but let’s give some consideration to Senator Jon Tester of Montana.
Tester is categorized as a Left-Liberal by Ballotopedia. He is currently the chair of the DSCC (so held in some regard by his compatriots). He is an organic farmer; he started his career as a music teacher.
I won’t go into a detailed description of his legislative career (you can find that at Ballotopedia) but he is strongly pro-choice, strongly supports green energy initiatives, supports the EPA, strongly supports making voting easier, for raising taxes on the wealthy, and against school vouchers for starters.
He is not perfect, of course. He has an A- rating from the NRA (but an F rating from another pro-gun group), voted against the Dream Act (but voted FOR continuing aid to sanctuary cities), and has a anti marijuana-legalization stance.
A little known fact about Tester: when Al Franken was going through the recount mishegas Tester lent his Chief of Staff to Franken for the duration of the recount. When Franken was declared the winner there was a celebration for Franken at the AFL-CIO headquaters in DC and Tester (and maybe Reid) were the only fellow Senators to show up.
By the way, the governor of Montana is also a Democrat, so we wouldn’t have the same replacement problem that we would have with a Warren or Brown VP pick.