Can I check if a player is online if they aren't in-game?

You have to use HTTPService, so make sure HttpEnabled for the HttpService is enabled, then you can use this code as a script in ServerScriptStorage to print the online status of player's IDs that you chat when you say "status/" then the ID.

local http = game:GetService("HttpService")

game.Players.PlayerAdded:connect(function(plr)
    plr.Chatted:connect(function(msg)
        if msg:lower():sub(1,7) == "status/" then
            local id = tonumber(msg:sub(8)) --Get the ID after the "/" in "status/"
            if id ~= nil then --If the ID doesn't return nil after being converted to a number then continue.
                local status = http:GetAsync("http://apiroblox.netne.net/GetUserStatus.php?id="..id) --Get online status of the player from the ID.
                local m = Instance.new("Message", game.Workspace)
                m.Text = "Status of player ID "..id..": "..tostring(status)
                wait(3)
                m:Destroy()
            end
        end
    end)
end) 

NOTE: This will print the entire status of the player, so if they're online, it tells you what the player is doing, same as how it appears on people's profiles.

HTTP Service. This would require you to make your own website that reflected off of the information on roblox because of the rules of using HTTP Service - cannot perform on the Roblox website.