A Kickstarter was formed by the cast in an attempt to raise $750,000 in 45 days so they could get a single animated half hour made. Apparently major organizations did not have the confidence in a bunch of nerdy voice actors playing dungeons and dragons for others' viewing pleasure as a financially safe bet. 88,887 backers pledged $11,385,449 to help bring this project to life.
The group name "Vox Machina" is a reference to the fact that all of the players are voice actors. The phrase translates from Latin as "voice machine."
Since Matthew Mercer was the campaign DM without an actual player character, he voices many characters, from villagers and guards, to Sylas Briarwood. One guard also looks like an animated version of Matt, leading many fans to play "Where's Mercer" during each episode.
Vox Machina notoriously had a very hard time opening doors during their adventurers and campaign. Scanlan, Pike, and Keyleth struggling to open the door in Episode 1 is a nod to this. The motif continues throughout season one, notably in Episode 7 when Percy, Vax, and Scanlan cannot get through a door without excessive force and spells; this actually happened in the streamed campaign with Scanlan actually taking damage from the door.
Based on campaign one of Critical Role (2015), which started as a home game of Dungeons and Dragons, and then ran on Geek and Sundry's twitch channel from 2015-2017. The campaign can be watched on youtube and consists of about 400 hours of game play.