The game

Kingmakers co-op and multiplayer

Kingmakers multiplayer is co-op: up to 4 players online, through the entire campaign, with each player leading an army of their own. It is online only, so there is no split screen, but you can play solo without a connection. There is no PvP planned for Early Access.

Fighting on a castle wall at sunset, with some soldiers highlighted in yellow and bodies across the walkway. — Fighting along a castle wall at sunset; some soldiers are outlined in yellow. Frame at 2:44 in the Gameplay Deep Dive, tinyBuildGAMES.

The short answers

PlayersUp to 4: you and three friends

Online or localOnline only

Split screenNo; co-op is online only

Solo offlineYes, solo does not need a connection

Join mid-gameDrop-in, drop-out through the whole campaign

ArmiesEach player commands their own

VehiclesNot shared between players, as first planned

PvPNot planned for Early Access; maybe later

Crossplay (Steam and Epic)Not announced; named only as work extra stores would add

How many players

4players at most, online

4 armiesone for each player

Whole campaigndrop-in, drop-out from start to finish

“4 players total can play together, so you and 3 friends!”
Official FAQ. From the Kingmakers Official FAQ (Steam discussions)

“We’ve designed Kingmakers to be a great single-player experience. But the entire campaign is playable in drop-in, drop-out four-player co-op.”
Redemption Road. Said in the tinyBuildGAMES video: Kingmakers - Gameplay Deep Dive

Everyone brings an army

Co-op here is not four players sharing one force. The Deep Dive says each player commands their own, and the Epic store page describes it the same way:

“Up to three friends can lead their own armies alongside yours, and join in on your conquest in online co-op.”
Epic Games Store page. From the Kingmakers Epic Games Store page

Those armies fight alongside yours, for the same Welsh rebellion; the sides are set out on Kingmakers factions. The recruits those armies are built from are on Kingmakers units and officers. One early limitation from the developer blog: players would not share vehicles at first, each getting their own.

Split screen, local co-op and offline play

Together

Online only

“No, co-op mode will be online only.”
Official FAQ. From the Kingmakers Official FAQ (Steam discussions)

Alone

Works offline

“Yes, you will not require an online connection to play solo.”
Official FAQ. From the Kingmakers Official FAQ (Steam discussions)

So playing together in one room still means one PC each, and a solo campaign works offline. The planned demo is also described as fully multiplayer; see the Kingmakers demo.

PvP

“It’s not in the plans for Early Access release, but we may consider it later for our roadmap if it’s highly requested.”
Official FAQ. From the Kingmakers Official FAQ (Steam discussions)

A later developer blog listed “maybe even a PvP mode” among things wanted after Early Access. That is the whole record.

Crossplay

Not announced, for or against. It has come up once in the game director’s replies, and only sideways: asked whether the game would be sold on GOG, he listed crossplay among the work every extra storefront adds.

Paul Fisch Game director, replying on the Steam forum

“GoG isn’t ruled out or anything. For EA it isn’t targeted right now though because getting on additional storefronts is actually a significant amount of work in terms of supporting crossplay/achievements/etc, and we are focused on getting to EA as soon as we can.”

Thread: Paul Fisch on GOG and other storefronts, Steam discussions

That says crossplay is part of the cost of selling in more places. It does not say whether Steam and Epic players will be able to play together. No plan for crossplay with PS5 or Xbox has been announced either; the console plans are on Kingmakers on consoles, and what each store page lists is on Kingmakers crossplay.